9 10

9_9

9_11

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 10's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 10 at Knotilus!

Knot presentations

Planar diagram presentation	X8291 X12,4,13,3 X18,10,1,9 X10,18,11,17 X16,8,17,7 X2,12,3,11 X4,16,5,15 X14,6,15,5 X6,14,7,13
Gauss code	1, -6, 2, -7, 8, -9, 5, -1, 3, -4, 6, -2, 9, -8, 7, -5, 4, -3
Dowker-Thistlethwaite code	8 12 14 16 18 2 6 4 10
Conway Notation	[333]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 11, width is 4, Braid index is 4		[{3, 11}, {2, 4}, {5, 3}, {4, 10}, {1, 5}, {11, 9}, {10, 6}, {7, 2}, {6, 8}, {9, 7}, {8, 1}]

[edit Notes on presentations of 9 10]

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 10"];

In[4]:=

PD[K]

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

Out[4]=

X8291 X12,4,13,3 X18,10,1,9 X10,18,11,17 X16,8,17,7 X2,12,3,11 X4,16,5,15 X14,6,15,5 X6,14,7,13

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

8 12 14 16 18 2 6 4 10

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[333]

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,-1,2,2,2,2,3,-2,3\})}

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[{3, 11}, {2, 4}, {5, 3}, {4, 10}, {1, 5}, {11, 9}, {10, 6}, {7, 2}, {6, 8}, {9, 7}, {8, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 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 \{2,3\}} 3-genus 2 Bridge index 2 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 \{4,6\}} Nakanishi index 1 Maximal Thurston-Bennequin number [3][-14] Hyperbolic Volume 8.77346 A-Polynomial See Data:9 10/A-polynomial

[edit Notes for 9 10's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ 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 10'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 4 t^2-8 t+9-8 t^{-1} +4 t^{-2} }
Conway polynomial	$4z^{4}+8z^{2}+1$
2nd Alexander ideal (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 \{1\}}
Determinant and Signature	{ 33, 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 -q^{11}+q^{10}-3 q^9+5 q^8-5 q^7+6 q^6-5 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} +z^4 a^{-8} +2 z^2 a^{-4} +5 z^2 a^{-6} +2 z^2 a^{-8} -z^2 a^{-10} +2 a^{-6} + a^{-8} -2 a^{-10} }
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^8 a^{-8} +z^8 a^{-10} +2 z^7 a^{-7} +3 z^7 a^{-9} +z^7 a^{-11} +3 z^6 a^{-6} -z^6 a^{-8} -3 z^6 a^{-10} +z^6 a^{-12} +2 z^5 a^{-5} -3 z^5 a^{-7} -7 z^5 a^{-9} -z^5 a^{-11} +z^5 a^{-13} +z^4 a^{-4} -7 z^4 a^{-6} +3 z^4 a^{-8} +9 z^4 a^{-10} -2 z^4 a^{-12} -3 z^3 a^{-5} +3 z^3 a^{-7} +9 z^3 a^{-9} -z^3 a^{-11} -4 z^3 a^{-13} -2 z^2 a^{-4} +7 z^2 a^{-6} -2 z^2 a^{-8} -11 z^2 a^{-10} -4 z a^{-9} +4 z a^{-13} -2 a^{-6} + a^{-8} +2 a^{-10} }
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} +2 q^{-16} +2 q^{-20} + q^{-22} + q^{-24} + q^{-26} -2 q^{-28} - q^{-30} - q^{-32} - q^{-34} }
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} +7 q^{-44} -9 q^{-46} +11 q^{-48} -8 q^{-50} +3 q^{-52} +5 q^{-54} -13 q^{-56} +21 q^{-58} -19 q^{-60} +12 q^{-62} -2 q^{-64} -10 q^{-66} +18 q^{-68} -17 q^{-70} +14 q^{-72} -2 q^{-74} -7 q^{-76} +13 q^{-78} -9 q^{-80} - q^{-82} +12 q^{-84} -19 q^{-86} +18 q^{-88} -9 q^{-90} -4 q^{-92} +21 q^{-94} -28 q^{-96} +31 q^{-98} -20 q^{-100} +6 q^{-102} +11 q^{-104} -21 q^{-106} +26 q^{-108} -18 q^{-110} +12 q^{-112} +2 q^{-114} -10 q^{-116} +15 q^{-118} -10 q^{-120} -2 q^{-122} +9 q^{-124} -16 q^{-126} +10 q^{-128} -3 q^{-130} -12 q^{-132} +18 q^{-134} -20 q^{-136} +15 q^{-138} -10 q^{-140} -9 q^{-142} +13 q^{-144} -16 q^{-146} +13 q^{-148} -9 q^{-150} +2 q^{-152} +3 q^{-154} -4 q^{-156} +6 q^{-158} -6 q^{-160} +4 q^{-162} - q^{-164} + q^{-168} -2 q^{-170} +2 q^{-172} + q^{-176} }

Further Quantum Invariants

Further quantum knot invariants for 9_10.

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^{-9} + q^{-11} + q^{-13} +2 q^{-17} -2 q^{-19} - q^{-23} }
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} +4 q^{-12} -2 q^{-14} -3 q^{-16} +7 q^{-18} - q^{-20} -6 q^{-22} +7 q^{-24} + q^{-26} -5 q^{-28} +2 q^{-30} +2 q^{-32} -3 q^{-36} +4 q^{-38} +3 q^{-40} -7 q^{-42} + q^{-44} +5 q^{-46} -7 q^{-48} -2 q^{-50} +4 q^{-52} -3 q^{-54} - q^{-56} +2 q^{-58} + q^{-64} }
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} +2 q^{-15} +2 q^{-17} -2 q^{-19} -3 q^{-21} +4 q^{-23} +7 q^{-25} -4 q^{-27} -10 q^{-29} +2 q^{-31} +17 q^{-33} +2 q^{-35} -20 q^{-37} -6 q^{-39} +21 q^{-41} +12 q^{-43} -20 q^{-45} -14 q^{-47} +17 q^{-49} +15 q^{-51} -9 q^{-53} -14 q^{-55} +3 q^{-57} +9 q^{-59} +3 q^{-61} -9 q^{-63} -9 q^{-65} +6 q^{-67} +15 q^{-69} -2 q^{-71} -20 q^{-73} + q^{-75} +20 q^{-77} +3 q^{-79} -23 q^{-81} -9 q^{-83} +16 q^{-85} +13 q^{-87} -16 q^{-89} -13 q^{-91} +8 q^{-93} +14 q^{-95} -2 q^{-97} -10 q^{-99} + q^{-101} +7 q^{-103} +2 q^{-105} -3 q^{-107} + q^{-111} + q^{-113} - q^{-115} - q^{-123} }
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^{-12} - q^{-14} +2 q^{-18} +2 q^{-22} -3 q^{-24} - q^{-26} +5 q^{-28} +5 q^{-32} -8 q^{-34} -6 q^{-36} +9 q^{-38} +6 q^{-40} +14 q^{-42} -17 q^{-44} -24 q^{-46} + q^{-48} +19 q^{-50} +47 q^{-52} -10 q^{-54} -51 q^{-56} -33 q^{-58} +14 q^{-60} +83 q^{-62} +23 q^{-64} -54 q^{-66} -70 q^{-68} -17 q^{-70} +90 q^{-72} +55 q^{-74} -28 q^{-76} -73 q^{-78} -42 q^{-80} +56 q^{-82} +55 q^{-84} +6 q^{-86} -43 q^{-88} -44 q^{-90} +11 q^{-92} +35 q^{-94} +26 q^{-96} -10 q^{-98} -34 q^{-100} -28 q^{-102} +15 q^{-104} +44 q^{-106} +17 q^{-108} -28 q^{-110} -58 q^{-112} +60 q^{-116} +42 q^{-118} -19 q^{-120} -84 q^{-122} -19 q^{-124} +59 q^{-126} +62 q^{-128} +5 q^{-130} -86 q^{-132} -45 q^{-134} +27 q^{-136} +66 q^{-138} +43 q^{-140} -53 q^{-142} -52 q^{-144} -10 q^{-146} +38 q^{-148} +55 q^{-150} -8 q^{-152} -29 q^{-154} -28 q^{-156} +2 q^{-158} +32 q^{-160} +9 q^{-162} -4 q^{-164} -16 q^{-166} -9 q^{-168} +8 q^{-170} +3 q^{-172} +4 q^{-174} -3 q^{-176} -4 q^{-178} + q^{-180} -2 q^{-182} +2 q^{-184} - q^{-188} + q^{-190} - q^{-192} + q^{-200} }
5	$q^{-15}-q^{-17}+2q^{-21}+q^{-27}-q^{-29}-q^{-31}+3q^{-33}+3q^{-35}-2q^{-37}-q^{-41}+4q^{-45}+6q^{-47}-q^{-49}-9q^{-51}-11q^{-53}-4q^{-55}+14q^{-57}+26q^{-59}+22q^{-61}-9q^{-63}-49q^{-65}-52q^{-67}-8q^{-69}+62q^{-71}+99q^{-73}+56q^{-75}-60q^{-77}-153q^{-79}-118q^{-81}+31q^{-83}+186q^{-85}+199q^{-87}+30q^{-89}-197q^{-91}-273q^{-93}-104q^{-95}+171q^{-97}+314q^{-99}+183q^{-101}-115q^{-103}-323q^{-105}-243q^{-107}+53q^{-109}+287q^{-111}+266q^{-113}+15q^{-115}-227q^{-117}-259q^{-119}-71q^{-121}+158q^{-123}+220q^{-125}+96q^{-127}-80q^{-129}-173q^{-131}-115q^{-133}+26q^{-135}+121q^{-137}+115q^{-139}+26q^{-141}-78q^{-143}-122q^{-145}-65q^{-147}+45q^{-149}+124q^{-151}+100q^{-153}-22q^{-155}-140q^{-157}-139q^{-159}+5q^{-161}+164q^{-163}+177q^{-165}+17q^{-167}-179q^{-169}-224q^{-171}-48q^{-173}+193q^{-175}+264q^{-177}+89q^{-179}-176q^{-181}-302q^{-183}-140q^{-185}+144q^{-187}+300q^{-189}+204q^{-191}-76q^{-193}-287q^{-195}-239q^{-197}+2q^{-199}+221q^{-201}+257q^{-203}+86q^{-205}-151q^{-207}-235q^{-209}-128q^{-211}+58q^{-213}+184q^{-215}+157q^{-217}+16q^{-219}-118q^{-221}-140q^{-223}-62q^{-225}+48q^{-227}+102q^{-229}+76q^{-231}-3q^{-233}-64q^{-235}-64q^{-237}-23q^{-239}+22q^{-241}+44q^{-243}+25q^{-245}-3q^{-247}-20q^{-249}-20q^{-251}-7q^{-253}+9q^{-255}+11q^{-257}+6q^{-259}+2q^{-261}-5q^{-263}-6q^{-265}+q^{-269}+q^{-271}+4q^{-273}-2q^{-277}-q^{-283}+q^{-285}+q^{-287}-q^{-295}$

A2 Invariants.

Weight	Invariant
1,0	$q^{-6}-q^{-8}+q^{-10}+2q^{-16}+2q^{-20}+q^{-22}+q^{-24}+q^{-26}-2q^{-28}-q^{-30}-q^{-32}-q^{-34}$
1,1	$q^{-12}-2q^{-14}+4q^{-16}-8q^{-18}+17q^{-20}-22q^{-22}+32q^{-24}-42q^{-26}+58q^{-28}-56q^{-30}+60q^{-32}-54q^{-34}+43q^{-36}-16q^{-38}-12q^{-40}+44q^{-42}-69q^{-44}+100q^{-46}-114q^{-48}+118q^{-50}-119q^{-52}+94q^{-54}-80q^{-56}+46q^{-58}-24q^{-60}-8q^{-62}+36q^{-64}-46q^{-66}+56q^{-68}-56q^{-70}+56q^{-72}-44q^{-74}+30q^{-76}-28q^{-78}+16q^{-80}-12q^{-82}+8q^{-84}-4q^{-86}+4q^{-88}+q^{-92}$
2,0	$q^{-12}-q^{-14}-q^{-16}+3q^{-18}+2q^{-20}-3q^{-22}-q^{-24}+4q^{-26}+3q^{-28}-2q^{-30}+q^{-32}+4q^{-34}-q^{-36}-2q^{-38}+2q^{-40}+q^{-42}+4q^{-46}+4q^{-48}+2q^{-54}-6q^{-58}-q^{-60}-3q^{-64}-6q^{-66}-3q^{-68}-q^{-72}-q^{-74}+2q^{-78}+q^{-80}+2q^{-82}+q^{-84}+q^{-86}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{-12}-q^{-14}+2q^{-18}-3q^{-20}+q^{-22}+6q^{-24}-4q^{-26}+q^{-28}+7q^{-30}-2q^{-32}+q^{-34}+6q^{-36}+q^{-38}+2q^{-44}-2q^{-46}-4q^{-48}+3q^{-50}-8q^{-54}+2q^{-56}-q^{-58}-7q^{-60}+q^{-62}-q^{-66}+2q^{-68}+2q^{-70}+q^{-74}$
1,0,0	$q^{-9}-q^{-11}+q^{-13}-q^{-15}+q^{-17}+2q^{-21}+q^{-23}+q^{-25}+2q^{-27}+q^{-29}+2q^{-31}+q^{-35}-2q^{-37}-q^{-39}-2q^{-41}-q^{-43}-q^{-45}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-18}-q^{-20}+q^{-24}-q^{-26}-q^{-28}+3q^{-30}+2q^{-32}-2q^{-34}+q^{-36}+5q^{-38}+2q^{-40}-2q^{-42}+4q^{-44}+8q^{-46}+q^{-50}+6q^{-52}+3q^{-54}-q^{-56}+4q^{-58}+5q^{-60}-q^{-64}+2q^{-66}-4q^{-68}-11q^{-70}-7q^{-72}-5q^{-74}-10q^{-76}-8q^{-78}+2q^{-82}+q^{-84}+2q^{-86}+5q^{-88}+3q^{-90}+q^{-92}+q^{-94}+q^{-96}$
1,0,0,0	$q^{-12}-q^{-14}+q^{-16}-q^{-18}+q^{-22}+2q^{-26}+q^{-28}+2q^{-30}+q^{-32}+2q^{-34}+q^{-36}+2q^{-38}+q^{-40}+q^{-44}-2q^{-46}-q^{-48}-2q^{-50}-2q^{-52}-q^{-54}-q^{-56}$

B2 Invariants.

Weight	Invariant
0,1	$q^{-12}-q^{-14}+2q^{-16}-4q^{-18}+5q^{-20}-5q^{-22}+6q^{-24}-4q^{-26}+5q^{-28}-q^{-30}+5q^{-34}-6q^{-36}+9q^{-38}-10q^{-40}+10q^{-42}-10q^{-44}+8q^{-46}-6q^{-48}+3q^{-50}-2q^{-54}+4q^{-56}-5q^{-58}+5q^{-60}-5q^{-62}+4q^{-64}-3q^{-66}+2q^{-68}-2q^{-70}-q^{-74}$
1,0	$q^{-18}-q^{-22}-q^{-24}+q^{-26}+3q^{-28}-4q^{-32}-2q^{-34}+4q^{-36}+6q^{-38}-q^{-40}-5q^{-42}-2q^{-44}+6q^{-46}+5q^{-48}-2q^{-50}-4q^{-52}+3q^{-54}+5q^{-56}+q^{-58}-3q^{-60}+4q^{-64}+2q^{-66}-3q^{-68}-3q^{-70}+2q^{-72}+3q^{-74}-2q^{-76}-4q^{-78}+q^{-80}+5q^{-82}-6q^{-86}-5q^{-88}+3q^{-90}+4q^{-92}-3q^{-94}-7q^{-96}-3q^{-98}+3q^{-100}+2q^{-102}-q^{-104}-2q^{-106}+2q^{-110}+2q^{-112}+q^{-120}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-18}-q^{-20}+q^{-22}-2q^{-24}+3q^{-26}-4q^{-28}+4q^{-30}-3q^{-32}+6q^{-34}-3q^{-36}+4q^{-38}-q^{-40}+5q^{-42}+2q^{-44}-q^{-46}+5q^{-48}-2q^{-50}+9q^{-52}-5q^{-54}+8q^{-56}-7q^{-58}+9q^{-60}-7q^{-62}+5q^{-64}-8q^{-66}+2q^{-68}-3q^{-70}-q^{-72}-2q^{-74}-4q^{-76}+2q^{-78}-5q^{-80}+2q^{-82}-6q^{-84}+3q^{-86}-4q^{-88}+2q^{-90}-2q^{-92}+3q^{-94}+2q^{-98}+q^{-102}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-30}-q^{-32}+2q^{-34}-3q^{-36}+2q^{-38}-q^{-40}-2q^{-42}+7q^{-44}-9q^{-46}+11q^{-48}-8q^{-50}+3q^{-52}+5q^{-54}-13q^{-56}+21q^{-58}-19q^{-60}+12q^{-62}-2q^{-64}-10q^{-66}+18q^{-68}-17q^{-70}+14q^{-72}-2q^{-74}-7q^{-76}+13q^{-78}-9q^{-80}-q^{-82}+12q^{-84}-19q^{-86}+18q^{-88}-9q^{-90}-4q^{-92}+21q^{-94}-28q^{-96}+31q^{-98}-20q^{-100}+6q^{-102}+11q^{-104}-21q^{-106}+26q^{-108}-18q^{-110}+12q^{-112}+2q^{-114}-10q^{-116}+15q^{-118}-10q^{-120}-2q^{-122}+9q^{-124}-16q^{-126}+10q^{-128}-3q^{-130}-12q^{-132}+18q^{-134}-20q^{-136}+15q^{-138}-10q^{-140}-9q^{-142}+13q^{-144}-16q^{-146}+13q^{-148}-9q^{-150}+2q^{-152}+3q^{-154}-4q^{-156}+6q^{-158}-6q^{-160}+4q^{-162}-q^{-164}+q^{-168}-2q^{-170}+2q^{-172}+q^{-176}$

.

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 10"];

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 4 t^2-8 t+9-8 t^{-1} +4 t^{-2} }

In[5]:=

Conway[K][z]

Out[5]=

$4z^{4}+8z^{2}+1$

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]=

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 \{1\}}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 33, 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 -q^{11}+q^{10}-3 q^9+5 q^8-5 q^7+6 q^6-5 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} +z^4 a^{-8} +2 z^2 a^{-4} +5 z^2 a^{-6} +2 z^2 a^{-8} -z^2 a^{-10} +2 a^{-6} + a^{-8} -2 a^{-10} }

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^8 a^{-8} +z^8 a^{-10} +2 z^7 a^{-7} +3 z^7 a^{-9} +z^7 a^{-11} +3 z^6 a^{-6} -z^6 a^{-8} -3 z^6 a^{-10} +z^6 a^{-12} +2 z^5 a^{-5} -3 z^5 a^{-7} -7 z^5 a^{-9} -z^5 a^{-11} +z^5 a^{-13} +z^4 a^{-4} -7 z^4 a^{-6} +3 z^4 a^{-8} +9 z^4 a^{-10} -2 z^4 a^{-12} -3 z^3 a^{-5} +3 z^3 a^{-7} +9 z^3 a^{-9} -z^3 a^{-11} -4 z^3 a^{-13} -2 z^2 a^{-4} +7 z^2 a^{-6} -2 z^2 a^{-8} -11 z^2 a^{-10} -4 z a^{-9} +4 z a^{-13} -2 a^{-6} + a^{-8} +2 a^{-10} }

"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 10"];

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]=

{ $4t^{2}-8t+9-8t^{-1}+4t^{-2}$, $-q^{11}+q^{10}-3q^{9}+5q^{8}-5q^{7}+6q^{6}-5q^{5}+4q^{4}-2q^{3}+q^{2}$ }

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:

(8, 22)

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 $32$ $176$ $512$ ${\frac {3856}{3}}$ 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{656}{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 5632} 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{30752}{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{5504}{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 1520} 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{16384}{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 15488} ${\frac {123392}{3}}$ ${\frac {20992}{3}}$ 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{1241164}{15}} 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{6064}{15}} 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{1589776}{45}} 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{5108}{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 \frac{72844}{15}}

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 10. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 χ 23                   1 -1 21                     0 19               3 1   -2 17             2       2 15           3 3       0 13         3 2         1 11       2 3           1 9     2 3             -1 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}^{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=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^{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=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}^{2}\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}^{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=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^{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=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 {\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=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 {\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}}

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^{31}-q^{30}+3 q^{28}-4 q^{27}-2 q^{26}+10 q^{25}-10 q^{24}-7 q^{23}+22 q^{22}-14 q^{21}-15 q^{20}+32 q^{19}-13 q^{18}-22 q^{17}+35 q^{16}-11 q^{15}-22 q^{14}+28 q^{13}-5 q^{12}-16 q^{11}+15 q^{10}-8 q^8+5 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 -q^{60}+q^{59}-2 q^{56}+3 q^{55}-q^{53}-5 q^{52}+8 q^{51}+5 q^{50}-7 q^{49}-16 q^{48}+16 q^{47}+21 q^{46}-13 q^{45}-37 q^{44}+13 q^{43}+50 q^{42}-10 q^{41}-62 q^{40}-q^{39}+76 q^{38}+7 q^{37}-81 q^{36}-22 q^{35}+94 q^{34}+24 q^{33}-90 q^{32}-37 q^{31}+94 q^{30}+36 q^{29}-84 q^{28}-43 q^{27}+77 q^{26}+41 q^{25}-60 q^{24}-41 q^{23}+46 q^{22}+35 q^{21}-28 q^{20}-32 q^{19}+19 q^{18}+21 q^{17}-6 q^{16}-17 q^{15}+4 q^{14}+9 q^{13}-6 q^{11}+q^{10}+2 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^{98}-q^{97}-q^{94}+3 q^{93}-3 q^{92}+q^{91}+2 q^{90}-5 q^{89}+6 q^{88}-8 q^{87}+2 q^{86}+9 q^{85}-6 q^{84}+11 q^{83}-25 q^{82}-5 q^{81}+21 q^{80}+7 q^{79}+34 q^{78}-55 q^{77}-35 q^{76}+20 q^{75}+28 q^{74}+97 q^{73}-72 q^{72}-83 q^{71}-22 q^{70}+27 q^{69}+193 q^{68}-49 q^{67}-122 q^{66}-94 q^{65}-14 q^{64}+284 q^{63}+8 q^{62}-125 q^{61}-172 q^{60}-79 q^{59}+349 q^{58}+69 q^{57}-107 q^{56}-232 q^{55}-137 q^{54}+379 q^{53}+114 q^{52}-80 q^{51}-261 q^{50}-180 q^{49}+373 q^{48}+138 q^{47}-44 q^{46}-252 q^{45}-204 q^{44}+318 q^{43}+139 q^{42}+5 q^{41}-203 q^{40}-203 q^{39}+220 q^{38}+108 q^{37}+50 q^{36}-120 q^{35}-168 q^{34}+113 q^{33}+55 q^{32}+66 q^{31}-43 q^{30}-108 q^{29}+44 q^{28}+8 q^{27}+48 q^{26}-2 q^{25}-51 q^{24}+16 q^{23}-10 q^{22}+23 q^{21}+5 q^{20}-20 q^{19}+8 q^{18}-7 q^{17}+8 q^{16}+3 q^{15}-7 q^{14}+3 q^{13}-2 q^{12}+2 q^{11}+q^{10}-2 q^9+q^8}

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_9 9_11