9 49

9_48

10_1

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 49's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 49 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X12,8,13,7 X5,15,6,14 X3,11,4,10 X11,3,12,2 X15,5,16,4 X17,9,18,8 X9,17,10,16 X18,14,1,13
Gauss code	1, 5, -4, 6, -3, -1, 2, 7, -8, 4, -5, -2, 9, 3, -6, 8, -7, -9
Dowker-Thistlethwaite code	6 -10 -14 12 -16 -2 18 -4 -8
Conway Notation	[-20:-20:-20]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 11, width is 4, Braid index is 4		[{2, 7}, {1, 5}, {8, 3}, {7, 9}, {6, 2}, {4, 1}, {5, 8}, {3, 6}, {9, 4}]

[edit Notes on presentations of 9 49]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["9 49"];

In[4]:=

PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

X6271 X12,8,13,7 X5,15,6,14 X3,11,4,10 X11,3,12,2 X15,5,16,4 X17,9,18,8 X9,17,10,16 X18,14,1,13

In[5]:=

GaussCode[K]

Out[5]=

1, 5, -4, 6, -3, -1, 2, 7, -8, 4, -5, -2, 9, 3, -6, 8, -7, -9

In[6]:=

DTCode[K]

Out[6]=

6 -10 -14 12 -16 -2 18 -4 -8

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[-20:-20:-20]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{BR}(4,\{1,1,2,1,1,-3,2,-1,2,3,3\})}

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]=

{ 4, 11, 4 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{2, 7}, {1, 5}, {8, 3}, {7, 9}, {6, 2}, {4, 1}, {5, 8}, {3, 6}, {9, 4}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 2 Bridge index 3 Super bridge index Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{4,5\}} Nakanishi index 2 Maximal Thurston-Bennequin number [3][-12] Hyperbolic Volume 9.42707 A-Polynomial See Data:9 49/A-polynomial

[edit Notes for 9 49's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} Topological 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} Concordance genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} Rasmussen s-Invariant -4

[edit Notes for 9 49's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 t^2-6 t+7-6 t^{-1} +3 t^{-2} }
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 z^4+6 z^2+1}
2nd Alexander ideal (db, data sources)	$\{5,t+1\}$
Determinant and Signature	{ 25, 4 }
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 q^9+3 q^8-4 q^7+5 q^6-4 q^5+4 q^4-2 q^3+q^2}
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^4 a^{-4} +2 z^4 a^{-6} +2 z^2 a^{-4} +6 z^2 a^{-6} -2 z^2 a^{-8} +4 a^{-6} -3 a^{-8} }
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^7 a^{-7} +z^7 a^{-9} +3 z^6 a^{-6} +4 z^6 a^{-8} +z^6 a^{-10} +2 z^5 a^{-5} +z^5 a^{-7} -z^5 a^{-9} +z^4 a^{-4} -8 z^4 a^{-6} -9 z^4 a^{-8} -3 z^3 a^{-5} -3 z^3 a^{-7} +3 z^3 a^{-9} +3 z^3 a^{-11} -2 z^2 a^{-4} +9 z^2 a^{-6} +10 z^2 a^{-8} -z^2 a^{-10} +2 z a^{-7} -2 z a^{-9} -4 z a^{-11} -4 a^{-6} -3 a^{-8} }
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-6} - q^{-8} + q^{-10} + q^{-14} +3 q^{-16} + q^{-18} +2 q^{-20} - q^{-22} - q^{-24} - q^{-26} -2 q^{-28} }
The G2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-30} - q^{-32} +2 q^{-34} -3 q^{-36} +2 q^{-38} - q^{-40} -2 q^{-42} +8 q^{-44} -10 q^{-46} +12 q^{-48} -7 q^{-50} - q^{-52} +10 q^{-54} -16 q^{-56} +19 q^{-58} -11 q^{-60} + q^{-62} +10 q^{-64} -14 q^{-66} +13 q^{-68} -2 q^{-70} -6 q^{-72} +14 q^{-74} -12 q^{-76} +4 q^{-78} +9 q^{-80} -15 q^{-82} +21 q^{-84} -16 q^{-86} +9 q^{-88} +5 q^{-90} -13 q^{-92} +22 q^{-94} -22 q^{-96} +16 q^{-98} -4 q^{-100} -7 q^{-102} +13 q^{-104} -16 q^{-106} +9 q^{-108} - q^{-110} -10 q^{-112} +10 q^{-114} -11 q^{-116} -2 q^{-118} +10 q^{-120} -20 q^{-122} +16 q^{-124} -9 q^{-126} -4 q^{-128} +11 q^{-130} -16 q^{-132} +15 q^{-134} -7 q^{-136} + q^{-138} +3 q^{-140} -7 q^{-142} +7 q^{-144} -2 q^{-146} + q^{-148} + q^{-150} }

Further Quantum Invariants

Further quantum knot invariants for 9_49.

A1 Invariants.

Weight	Invariant
1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-3} - q^{-5} +2 q^{-7} + q^{-11} + q^{-13} - q^{-15} + q^{-17} -2 q^{-19} }
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-6} - q^{-8} +5 q^{-12} - q^{-14} -5 q^{-16} +5 q^{-18} +2 q^{-20} -5 q^{-22} +4 q^{-24} +4 q^{-26} -3 q^{-28} - q^{-30} + q^{-32} + q^{-34} -5 q^{-36} + q^{-38} +5 q^{-40} -6 q^{-42} -2 q^{-44} +5 q^{-46} -3 q^{-48} -3 q^{-50} +3 q^{-52} + q^{-54} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-9} - q^{-11} +3 q^{-15} +3 q^{-17} -3 q^{-19} -7 q^{-21} +3 q^{-23} +14 q^{-25} +4 q^{-27} -14 q^{-29} -13 q^{-31} +14 q^{-33} +18 q^{-35} -9 q^{-37} -21 q^{-39} +4 q^{-41} +23 q^{-43} + q^{-45} -19 q^{-47} -3 q^{-49} +14 q^{-51} +6 q^{-53} -9 q^{-55} -9 q^{-57} +2 q^{-59} +8 q^{-61} + q^{-63} -14 q^{-65} -7 q^{-67} +14 q^{-69} +14 q^{-71} -16 q^{-73} -17 q^{-75} +12 q^{-77} +21 q^{-79} -6 q^{-81} -21 q^{-83} - q^{-85} +18 q^{-87} +5 q^{-89} -10 q^{-91} -7 q^{-93} +5 q^{-95} +8 q^{-97} - q^{-99} -2 q^{-101} -2 q^{-103} }
4	$q^{-12}-q^{-14}+3q^{-18}+q^{-20}+q^{-22}-6q^{-24}-4q^{-26}+8q^{-28}+10q^{-30}+14q^{-32}-12q^{-34}-28q^{-36}-13q^{-38}+12q^{-40}+51q^{-42}+23q^{-44}-29q^{-46}-58q^{-48}-37q^{-50}+56q^{-52}+76q^{-54}+19q^{-56}-66q^{-58}-93q^{-60}+10q^{-62}+84q^{-64}+69q^{-66}-28q^{-68}-97q^{-70}-28q^{-72}+51q^{-74}+71q^{-76}+4q^{-78}-62q^{-80}-38q^{-82}+14q^{-84}+45q^{-86}+17q^{-88}-25q^{-90}-34q^{-92}-9q^{-94}+25q^{-96}+32q^{-98}+7q^{-100}-43q^{-102}-38q^{-104}+13q^{-106}+56q^{-108}+43q^{-110}-49q^{-112}-73q^{-114}-13q^{-116}+71q^{-118}+87q^{-120}-25q^{-122}-88q^{-124}-57q^{-126}+39q^{-128}+99q^{-130}+24q^{-132}-50q^{-134}-75q^{-136}-17q^{-138}+58q^{-140}+45q^{-142}+7q^{-144}-37q^{-146}-35q^{-148}+5q^{-150}+19q^{-152}+20q^{-154}-q^{-156}-13q^{-158}-7q^{-160}-3q^{-162}+3q^{-164}+3q^{-166}+q^{-168}$
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-15} - q^{-17} +3 q^{-21} + q^{-23} - q^{-25} -2 q^{-27} -3 q^{-29} +9 q^{-33} +14 q^{-35} +3 q^{-37} -10 q^{-39} -24 q^{-41} -23 q^{-43} +40 q^{-47} +59 q^{-49} +34 q^{-51} -28 q^{-53} -92 q^{-55} -97 q^{-57} -27 q^{-59} +96 q^{-61} +171 q^{-63} +117 q^{-65} -44 q^{-67} -206 q^{-69} -231 q^{-71} -64 q^{-73} +197 q^{-75} +325 q^{-77} +194 q^{-79} -114 q^{-81} -363 q^{-83} -326 q^{-85} -8 q^{-87} +339 q^{-89} +402 q^{-91} +127 q^{-93} -257 q^{-95} -424 q^{-97} -219 q^{-99} +159 q^{-101} +387 q^{-103} +263 q^{-105} -72 q^{-107} -310 q^{-109} -263 q^{-111} +5 q^{-113} +228 q^{-115} +229 q^{-117} +33 q^{-119} -156 q^{-121} -183 q^{-123} -56 q^{-125} +92 q^{-127} +141 q^{-129} +66 q^{-131} -52 q^{-133} -117 q^{-135} -81 q^{-137} +20 q^{-139} +108 q^{-141} +112 q^{-143} +16 q^{-145} -116 q^{-147} -156 q^{-149} -50 q^{-151} +121 q^{-153} +216 q^{-155} +116 q^{-157} -128 q^{-159} -281 q^{-161} -184 q^{-163} +100 q^{-165} +334 q^{-167} +279 q^{-169} -42 q^{-171} -353 q^{-173} -359 q^{-175} -46 q^{-177} +317 q^{-179} +417 q^{-181} +151 q^{-183} -229 q^{-185} -414 q^{-187} -254 q^{-189} +106 q^{-191} +349 q^{-193} +299 q^{-195} +25 q^{-197} -232 q^{-199} -292 q^{-201} -124 q^{-203} +107 q^{-205} +219 q^{-207} +160 q^{-209} +3 q^{-211} -120 q^{-213} -141 q^{-215} -61 q^{-217} +42 q^{-219} +82 q^{-221} +63 q^{-223} +12 q^{-225} -31 q^{-227} -46 q^{-229} -21 q^{-231} +3 q^{-233} +12 q^{-235} +17 q^{-237} +8 q^{-239} - q^{-241} -4 q^{-243} -2 q^{-245} -2 q^{-247} }

A2 Invariants.

Weight	Invariant
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-6} - q^{-8} + q^{-10} + q^{-14} +3 q^{-16} + q^{-18} +2 q^{-20} - q^{-22} - q^{-24} - q^{-26} -2 q^{-28} }
1,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-12} -2 q^{-14} +4 q^{-16} -8 q^{-18} +19 q^{-20} -22 q^{-22} +36 q^{-24} -42 q^{-26} +48 q^{-28} -40 q^{-30} +34 q^{-32} -10 q^{-34} -6 q^{-36} +38 q^{-38} -56 q^{-40} +70 q^{-42} -87 q^{-44} +78 q^{-46} -84 q^{-48} +58 q^{-50} -45 q^{-52} +18 q^{-54} +10 q^{-56} -22 q^{-58} +45 q^{-60} -48 q^{-62} +48 q^{-64} -42 q^{-66} +25 q^{-68} -22 q^{-70} +10 q^{-72} -2 q^{-74} +2 q^{-76} +2 q^{-78} }
2,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-12} - q^{-14} - q^{-16} +3 q^{-18} +3 q^{-20} - q^{-22} - q^{-24} +2 q^{-26} +2 q^{-28} -2 q^{-30} + q^{-32} +4 q^{-34} +2 q^{-36} +3 q^{-38} +5 q^{-40} +2 q^{-42} - q^{-44} - q^{-46} -2 q^{-48} -6 q^{-50} -4 q^{-52} - q^{-54} - q^{-56} -5 q^{-58} - q^{-60} + q^{-62} +3 q^{-70} + q^{-72} + q^{-74} }

A3 Invariants.

Weight	Invariant
0,1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-12} - q^{-14} +2 q^{-18} -3 q^{-20} +2 q^{-22} +6 q^{-24} -2 q^{-26} +5 q^{-28} +6 q^{-30} + q^{-34} +2 q^{-36} - q^{-38} -2 q^{-40} -3 q^{-42} -3 q^{-46} -6 q^{-48} +2 q^{-50} -2 q^{-52} -5 q^{-54} +4 q^{-56} + q^{-58} - q^{-60} +3 q^{-62} }
1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-9} - q^{-11} + q^{-13} - q^{-15} + q^{-17} + q^{-19} +3 q^{-21} +3 q^{-23} +2 q^{-25} +2 q^{-27} - q^{-29} - q^{-31} -3 q^{-33} - q^{-35} -2 q^{-37} }

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-18}-q^{-20}+q^{-24}-q^{-26}-q^{-28}+4q^{-30}+2q^{-32}-q^{-34}+4q^{-36}+7q^{-38}+3q^{-40}+2q^{-42}+8q^{-44}+8q^{-46}+3q^{-50}+5q^{-52}-3q^{-54}-6q^{-56}-2q^{-58}-8q^{-60}-13q^{-62}-8q^{-64}-5q^{-66}-6q^{-68}-3q^{-70}+6q^{-72}+6q^{-74}+2q^{-76}+3q^{-78}+4q^{-80}-q^{-84}$
1,0,0,0	$q^{-12}-q^{-14}+q^{-16}-q^{-18}+q^{-22}+q^{-24}+3q^{-26}+3q^{-28}+4q^{-30}+2q^{-32}+2q^{-34}-q^{-36}-q^{-38}-3q^{-40}-3q^{-42}-q^{-44}-2q^{-46}$

B2 Invariants.

Weight	Invariant
0,1	$q^{-12}-q^{-14}+2q^{-16}-4q^{-18}+5q^{-20}-4q^{-22}+4q^{-24}+3q^{-28}+2q^{-30}-2q^{-32}+7q^{-34}-8q^{-36}+7q^{-38}-8q^{-40}+5q^{-42}-6q^{-44}+3q^{-46}-2q^{-50}+4q^{-52}-3q^{-54}+4q^{-56}-5q^{-58}+3q^{-60}-3q^{-62}$
1,0	$q^{-18}-q^{-22}-q^{-24}+q^{-26}+3q^{-28}-4q^{-32}-q^{-34}+5q^{-36}+5q^{-38}-2q^{-40}-2q^{-42}+q^{-44}+6q^{-46}+2q^{-48}-2q^{-50}-q^{-52}+4q^{-54}+2q^{-56}-2q^{-58}-2q^{-60}+q^{-62}+3q^{-64}-q^{-66}-4q^{-68}-2q^{-70}+2q^{-72}-5q^{-76}-4q^{-78}+2q^{-80}+3q^{-82}-3q^{-84}-5q^{-86}-q^{-88}+4q^{-90}+3q^{-92}-2q^{-94}-2q^{-96}+q^{-98}+3q^{-100}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-18}-q^{-20}+q^{-22}-2q^{-24}+3q^{-26}-4q^{-28}+4q^{-30}-2q^{-32}+5q^{-34}+q^{-36}+4q^{-38}+4q^{-40}+5q^{-42}+6q^{-44}-2q^{-46}+6q^{-48}-5q^{-50}+5q^{-52}-8q^{-54}+2q^{-56}-8q^{-58}+3q^{-60}-5q^{-62}-4q^{-66}-2q^{-68}+q^{-70}-4q^{-72}+q^{-74}-4q^{-76}+4q^{-78}-2q^{-80}+4q^{-82}-2q^{-84}+3q^{-86}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-30}-q^{-32}+2q^{-34}-3q^{-36}+2q^{-38}-q^{-40}-2q^{-42}+8q^{-44}-10q^{-46}+12q^{-48}-7q^{-50}-q^{-52}+10q^{-54}-16q^{-56}+19q^{-58}-11q^{-60}+q^{-62}+10q^{-64}-14q^{-66}+13q^{-68}-2q^{-70}-6q^{-72}+14q^{-74}-12q^{-76}+4q^{-78}+9q^{-80}-15q^{-82}+21q^{-84}-16q^{-86}+9q^{-88}+5q^{-90}-13q^{-92}+22q^{-94}-22q^{-96}+16q^{-98}-4q^{-100}-7q^{-102}+13q^{-104}-16q^{-106}+9q^{-108}-q^{-110}-10q^{-112}+10q^{-114}-11q^{-116}-2q^{-118}+10q^{-120}-20q^{-122}+16q^{-124}-9q^{-126}-4q^{-128}+11q^{-130}-16q^{-132}+15q^{-134}-7q^{-136}+q^{-138}+3q^{-140}-7q^{-142}+7q^{-144}-2q^{-146}+q^{-148}+q^{-150}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 49"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 t^2-6 t+7-6 t^{-1} +3 t^{-2} }

In[5]:=

Conway[K][z]

Out[5]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 z^4+6 z^2+1}

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{5,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 25, 4 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 q^9+3 q^8-4 q^7+5 q^6-4 q^5+4 q^4-2 q^3+q^2}

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^4 a^{-4} +2 z^4 a^{-6} +2 z^2 a^{-4} +6 z^2 a^{-6} -2 z^2 a^{-8} +4 a^{-6} -3 a^{-8} }

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^7 a^{-7} +z^7 a^{-9} +3 z^6 a^{-6} +4 z^6 a^{-8} +z^6 a^{-10} +2 z^5 a^{-5} +z^5 a^{-7} -z^5 a^{-9} +z^4 a^{-4} -8 z^4 a^{-6} -9 z^4 a^{-8} -3 z^3 a^{-5} -3 z^3 a^{-7} +3 z^3 a^{-9} +3 z^3 a^{-11} -2 z^2 a^{-4} +9 z^2 a^{-6} +10 z^2 a^{-8} -z^2 a^{-10} +2 z a^{-7} -2 z a^{-9} -4 z a^{-11} -4 a^{-6} -3 a^{-8} }

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 49"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ $3t^{2}-6t+7-6t^{-1}+3t^{-2}$, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 q^9+3 q^8-4 q^7+5 q^6-4 q^5+4 q^4-2 q^3+q^2} }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(6, 14)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 24} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 112} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 288} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 700} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 116} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2688} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{14368}{3}} ${\frac {2560}{3}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 688} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2304} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6272} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 16800} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2784} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{166191}{5}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1876}{5}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{207244}{15}} ${\frac {689}{3}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{9391}{5}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 4 is the signature of 9 49. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 χ 19               2 -2 17             1   1 15           3 2   -1 13         2 1     1 11       2 3       1 9     2 2         0 7     2           2 5 1 2             -1 3 1               1

Integral Khovanov Homology (db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=6} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=7} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{26}+2 q^{25}-6 q^{24}+q^{23}+10 q^{22}-13 q^{21}-3 q^{20}+21 q^{19}-17 q^{18}-9 q^{17}+27 q^{16}-17 q^{15}-11 q^{14}+25 q^{13}-10 q^{12}-11 q^{11}+16 q^{10}-3 q^9-8 q^8+6 q^7+q^6-2 q^5+q^4}
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 q^{50}+q^{48}+9 q^{47}-5 q^{46}-12 q^{45}-2 q^{44}+24 q^{43}+8 q^{42}-31 q^{41}-22 q^{40}+39 q^{39}+35 q^{38}-40 q^{37}-51 q^{36}+40 q^{35}+65 q^{34}-40 q^{33}-72 q^{32}+33 q^{31}+80 q^{30}-33 q^{29}-78 q^{28}+22 q^{27}+80 q^{26}-18 q^{25}-70 q^{24}+5 q^{23}+64 q^{22}+2 q^{21}-48 q^{20}-14 q^{19}+39 q^{18}+14 q^{17}-21 q^{16}-18 q^{15}+12 q^{14}+13 q^{13}-3 q^{12}-8 q^{11}+q^{10}+3 q^9+q^8-2 q^7+q^6}
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{82}+2 q^{81}-6 q^{79}-4 q^{78}-5 q^{77}+14 q^{76}+21 q^{75}-7 q^{74}-18 q^{73}-45 q^{72}+12 q^{71}+65 q^{70}+31 q^{69}-5 q^{68}-120 q^{67}-46 q^{66}+90 q^{65}+105 q^{64}+70 q^{63}-180 q^{62}-142 q^{61}+59 q^{60}+168 q^{59}+182 q^{58}-196 q^{57}-226 q^{56}-q^{55}+192 q^{54}+274 q^{53}-183 q^{52}-269 q^{51}-52 q^{50}+187 q^{49}+324 q^{48}-158 q^{47}-276 q^{46}-86 q^{45}+162 q^{44}+333 q^{43}-116 q^{42}-248 q^{41}-117 q^{40}+110 q^{39}+309 q^{38}-50 q^{37}-181 q^{36}-137 q^{35}+31 q^{34}+240 q^{33}+19 q^{32}-84 q^{31}-122 q^{30}-43 q^{29}+137 q^{28}+46 q^{27}+q^{26}-65 q^{25}-63 q^{24}+44 q^{23}+25 q^{22}+30 q^{21}-13 q^{20}-35 q^{19}+5 q^{18}+15 q^{16}+3 q^{15}-9 q^{14}+q^{13}-2 q^{12}+3 q^{11}+q^{10}-2 q^9+q^8}
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{NotAvailable}(q)}
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{NotAvailable}(q)}
7	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{NotAvailable}(q)}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

9_48 10_1