9 15

9_14

9_16

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 15's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 15 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X7,10,8,11 X3948 X9,3,10,2 X13,17,14,16 X5,15,6,14 X15,7,16,6 X11,1,12,18 X17,13,18,12
Gauss code	-1, 4, -3, 1, -6, 7, -2, 3, -4, 2, -8, 9, -5, 6, -7, 5, -9, 8
Dowker-Thistlethwaite code	4 8 14 10 2 18 16 6 12
Conway Notation	[2322]

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

[edit Notes on presentations of 9 15]

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 15"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X7,10,8,11 X3948 X9,3,10,2 X13,17,14,16 X5,15,6,14 X15,7,16,6 X11,1,12,18 X17,13,18,12

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 14 10 2 18 16 6 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[2322]

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}(5,\{1,1,1,2,-1,-3,2,4,-3,4\})}

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]=

{ 5, 10, 5 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 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 \{4,5\}} Nakanishi index 1 Maximal Thurston-Bennequin number [-1][-10] Hyperbolic Volume 9.8855 A-Polynomial See Data:9 15/A-polynomial

[edit Notes for 9 15'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 1} 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 1} 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 2

[edit Notes for 9 15'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 -2 t^2+10 t-15+10 t^{-1} -2 t^{-2} }
Conway polynomial	$-2z^{4}+2z^{2}+1$
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	{ 39, 2 }
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^8+2 q^7-4 q^6+6 q^5-6 q^4+7 q^3-6 q^2+4 q-2+ q^{-1} }
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^{-2} -z^4 a^{-4} -z^2 a^{-2} +2 z^2 a^{-6} +z^2- a^{-2} + a^{-4} + a^{-6} - a^{-8} +1}
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^{-4} +z^8 a^{-6} +2 z^7 a^{-3} +4 z^7 a^{-5} +2 z^7 a^{-7} +2 z^6 a^{-2} +z^6 a^{-4} +z^6 a^{-6} +2 z^6 a^{-8} +2 z^5 a^{-1} -z^5 a^{-3} -7 z^5 a^{-5} -3 z^5 a^{-7} +z^5 a^{-9} -4 z^4 a^{-6} -5 z^4 a^{-8} +z^4-3 z^3 a^{-1} +5 z^3 a^{-5} -z^3 a^{-7} -3 z^3 a^{-9} -3 z^2 a^{-2} -2 z^2 a^{-4} +2 z^2 a^{-6} +3 z^2 a^{-8} -2 z^2+z a^{-1} +z a^{-3} -z a^{-5} +z a^{-7} +2 z a^{-9} + a^{-2} + a^{-4} - a^{-6} - a^{-8} +1}
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^4+2 q^{-2} -2 q^{-4} +2 q^{-12} +2 q^{-16} - q^{-20} + q^{-22} - q^{-24} - q^{-26} }
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^{18}-q^{16}+3 q^{14}-3 q^{12}+2 q^{10}-3 q^6+8 q^4-9 q^2+11-9 q^{-2} +5 q^{-4} +4 q^{-6} -13 q^{-8} +23 q^{-10} -26 q^{-12} +21 q^{-14} -13 q^{-16} -5 q^{-18} +20 q^{-20} -31 q^{-22} +33 q^{-24} -20 q^{-26} +2 q^{-28} +14 q^{-30} -23 q^{-32} +15 q^{-34} -2 q^{-36} -13 q^{-38} +24 q^{-40} -23 q^{-42} +9 q^{-44} +17 q^{-46} -34 q^{-48} +46 q^{-50} -42 q^{-52} +21 q^{-54} +6 q^{-56} -28 q^{-58} +43 q^{-60} -44 q^{-62} +36 q^{-64} -11 q^{-66} -11 q^{-68} +26 q^{-70} -30 q^{-72} +20 q^{-74} - q^{-76} -15 q^{-78} +21 q^{-80} -15 q^{-82} +3 q^{-84} +18 q^{-86} -31 q^{-88} +33 q^{-90} -23 q^{-92} +18 q^{-96} -32 q^{-98} +33 q^{-100} -24 q^{-102} +10 q^{-104} +3 q^{-106} -15 q^{-108} +17 q^{-110} -15 q^{-112} +9 q^{-114} -3 q^{-116} -2 q^{-118} +3 q^{-120} -4 q^{-122} +3 q^{-124} - q^{-126} + q^{-128} }

Further Quantum Invariants

Further quantum knot invariants for 9_15.

A1 Invariants.

Weight	Invariant
1	$q^{3}-q+2q^{-1}-2q^{-3}+q^{-5}+q^{-7}+2q^{-11}-2q^{-13}+q^{-15}-q^{-17}$
2	$q^{10}-q^{8}-q^{6}+3q^{4}-3q^{2}-2+9q^{-2}-4q^{-4}-6q^{-6}+11q^{-8}-q^{-10}-7q^{-12}+5q^{-14}+2q^{-16}-3q^{-18}-3q^{-20}+5q^{-22}+2q^{-24}-8q^{-26}+5q^{-28}+6q^{-30}-10q^{-32}+q^{-34}+7q^{-36}-6q^{-38}-2q^{-40}+4q^{-42}-q^{-44}-q^{-46}+q^{-48}$
3	$q^{21}-q^{19}-q^{17}+2q^{13}-q^{11}-2q^{9}+4q^{7}+4q^{5}-7q^{3}-8q+11q^{-1}+16q^{-3}-14q^{-5}-25q^{-7}+13q^{-9}+33q^{-11}-5q^{-13}-38q^{-15}+38q^{-19}+8q^{-21}-28q^{-23}-14q^{-25}+21q^{-27}+16q^{-29}-9q^{-31}-19q^{-33}-2q^{-35}+17q^{-37}+11q^{-39}-19q^{-41}-21q^{-43}+19q^{-45}+27q^{-47}-14q^{-49}-33q^{-51}+10q^{-53}+36q^{-55}-37q^{-59}-7q^{-61}+28q^{-63}+15q^{-65}-21q^{-67}-18q^{-69}+12q^{-71}+16q^{-73}-3q^{-75}-12q^{-77}-q^{-79}+7q^{-81}+2q^{-83}-3q^{-85}-q^{-87}+q^{-89}+q^{-91}-q^{-93}$
4	$q^{36}-q^{34}-q^{32}-q^{28}+4q^{26}-q^{24}+2q^{20}-5q^{18}+3q^{16}-7q^{14}+2q^{12}+15q^{10}-q^{8}-2q^{6}-32q^{4}-9q^{2}+41+34q^{-2}+13q^{-4}-78q^{-6}-62q^{-8}+46q^{-10}+92q^{-12}+74q^{-14}-95q^{-16}-135q^{-18}-7q^{-20}+113q^{-22}+151q^{-24}-49q^{-26}-159q^{-28}-75q^{-30}+68q^{-32}+162q^{-34}+20q^{-36}-103q^{-38}-97q^{-40}-2q^{-42}+107q^{-44}+62q^{-46}-25q^{-48}-78q^{-50}-50q^{-52}+33q^{-54}+76q^{-56}+40q^{-58}-56q^{-60}-85q^{-62}-25q^{-64}+91q^{-66}+95q^{-68}-35q^{-70}-115q^{-72}-81q^{-74}+96q^{-76}+143q^{-78}+5q^{-80}-118q^{-82}-136q^{-84}+55q^{-86}+152q^{-88}+67q^{-90}-67q^{-92}-161q^{-94}-20q^{-96}+101q^{-98}+99q^{-100}+15q^{-102}-113q^{-104}-64q^{-106}+17q^{-108}+70q^{-110}+61q^{-112}-37q^{-114}-47q^{-116}-28q^{-118}+16q^{-120}+44q^{-122}+4q^{-124}-9q^{-126}-21q^{-128}-7q^{-130}+14q^{-132}+4q^{-134}+3q^{-136}-5q^{-138}-4q^{-140}+3q^{-142}+q^{-146}-q^{-148}-q^{-150}+q^{-152}$
5	$q^{55}-q^{53}-q^{51}-q^{47}+q^{45}+4q^{43}+q^{41}-2q^{39}-5q^{35}-5q^{33}+3q^{31}+6q^{29}+7q^{27}+6q^{25}-5q^{23}-20q^{21}-19q^{19}+2q^{17}+30q^{15}+45q^{13}+21q^{11}-37q^{9}-88q^{7}-68q^{5}+34q^{3}+137q+146q^{-1}+8q^{-3}-182q^{-5}-260q^{-7}-97q^{-9}+206q^{-11}+382q^{-13}+235q^{-15}-169q^{-17}-487q^{-19}-410q^{-21}+67q^{-23}+553q^{-25}+581q^{-27}+84q^{-29}-529q^{-31}-704q^{-33}-266q^{-35}+436q^{-37}+765q^{-39}+412q^{-41}-287q^{-43}-721q^{-45}-510q^{-47}+116q^{-49}+604q^{-51}+547q^{-53}+28q^{-55}-451q^{-57}-501q^{-59}-150q^{-61}+277q^{-63}+434q^{-65}+222q^{-67}-136q^{-69}-336q^{-71}-267q^{-73}+5q^{-75}+263q^{-77}+298q^{-79}+91q^{-81}-198q^{-83}-334q^{-85}-181q^{-87}+156q^{-89}+385q^{-91}+269q^{-93}-125q^{-95}-443q^{-97}-366q^{-99}+79q^{-101}+496q^{-103}+476q^{-105}-16q^{-107}-530q^{-109}-576q^{-111}-88q^{-113}+522q^{-115}+667q^{-117}+208q^{-119}-441q^{-121}-711q^{-123}-352q^{-125}+317q^{-127}+695q^{-129}+462q^{-131}-137q^{-133}-595q^{-135}-548q^{-137}-43q^{-139}+449q^{-141}+534q^{-143}+195q^{-145}-252q^{-147}-461q^{-149}-292q^{-151}+79q^{-153}+328q^{-155}+302q^{-157}+61q^{-159}-178q^{-161}-252q^{-163}-137q^{-165}+57q^{-167}+168q^{-169}+141q^{-171}+26q^{-173}-80q^{-175}-110q^{-177}-59q^{-179}+21q^{-181}+63q^{-183}+54q^{-185}+9q^{-187}-25q^{-189}-34q^{-191}-19q^{-193}+7q^{-195}+18q^{-197}+11q^{-199}-4q^{-203}-7q^{-205}-3q^{-207}+4q^{-209}+2q^{-211}-q^{-213}-q^{-219}+q^{-221}+q^{-223}-q^{-225}$

A2 Invariants.

Weight	Invariant
1,0	$q^{4}+2q^{-2}-2q^{-4}+2q^{-12}+2q^{-16}-q^{-20}+q^{-22}-q^{-24}-q^{-26}$
1,1	$q^{12}-2q^{10}+6q^{8}-10q^{6}+17q^{4}-26q^{2}+36-52q^{-2}+68q^{-4}-82q^{-6}+98q^{-8}-102q^{-10}+106q^{-12}-82q^{-14}+52q^{-16}-8q^{-18}-47q^{-20}+102q^{-22}-156q^{-24}+194q^{-26}-217q^{-28}+220q^{-30}-204q^{-32}+174q^{-34}-126q^{-36}+72q^{-38}-12q^{-40}-38q^{-42}+78q^{-44}-108q^{-46}+118q^{-48}-116q^{-50}+98q^{-52}-80q^{-54}+58q^{-56}-38q^{-58}+23q^{-60}-12q^{-62}+6q^{-64}-2q^{-66}+q^{-68}$
2,0	$q^{12}-q^{8}+2q^{4}-4+7q^{-4}-q^{-6}-6q^{-8}+3q^{-10}+7q^{-12}+q^{-14}-4q^{-16}+3q^{-18}+3q^{-20}-3q^{-22}-q^{-24}-3q^{-28}-q^{-30}+5q^{-32}-q^{-34}-2q^{-36}+3q^{-38}+6q^{-40}-2q^{-42}-6q^{-44}+2q^{-46}+3q^{-48}-3q^{-50}-5q^{-52}+3q^{-56}-2q^{-60}+q^{-64}+q^{-66}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{8}-q^{6}+q^{4}+3q^{2}-3+6q^{-4}-6q^{-6}-2q^{-8}+9q^{-10}-5q^{-12}-2q^{-14}+6q^{-16}-2q^{-18}-2q^{-20}+q^{-22}+4q^{-24}-2q^{-28}+6q^{-30}+3q^{-32}-8q^{-34}+4q^{-36}+2q^{-38}-9q^{-40}+2q^{-42}+q^{-44}-4q^{-46}+2q^{-48}+q^{-50}-q^{-52}+q^{-54}$
1,0,0	$q^{5}+q+2q^{-3}-2q^{-5}-q^{-9}+q^{-15}+2q^{-17}+2q^{-21}+q^{-25}-q^{-27}+q^{-29}-q^{-31}-q^{-33}-q^{-35}$

B2 Invariants.

Weight	Invariant
0,1	$q^{8}-q^{6}+3q^{4}-3q^{2}+5-6q^{-2}+8q^{-4}-8q^{-6}+6q^{-8}-5q^{-10}+q^{-12}+2q^{-14}-6q^{-16}+10q^{-18}-12q^{-20}+15q^{-22}-14q^{-24}+14q^{-26}-10q^{-28}+8q^{-30}-3q^{-32}+4q^{-36}-6q^{-38}+7q^{-40}-8q^{-42}+7q^{-44}-6q^{-46}+4q^{-48}-3q^{-50}+q^{-52}-q^{-54}$
1,0	$q^{14}-q^{10}-q^{8}+2q^{6}+3q^{4}-4-3q^{-2}+3q^{-4}+7q^{-6}-8q^{-10}-4q^{-12}+6q^{-14}+8q^{-16}-2q^{-18}-7q^{-20}+7q^{-24}+2q^{-26}-6q^{-28}-3q^{-30}+4q^{-32}+4q^{-34}-3q^{-36}-4q^{-38}+2q^{-40}+6q^{-42}-4q^{-46}+7q^{-50}+3q^{-52}-6q^{-54}-7q^{-56}+4q^{-58}+8q^{-60}-q^{-62}-9q^{-64}-5q^{-66}+5q^{-68}+5q^{-70}-2q^{-72}-5q^{-74}-q^{-76}+3q^{-78}+2q^{-80}-q^{-82}-q^{-84}+q^{-88}$

G2 Invariants.

Weight

Invariant

1,0

$q^{18}-q^{16}+3q^{14}-3q^{12}+2q^{10}-3q^{6}+8q^{4}-9q^{2}+11-9q^{-2}+5q^{-4}+4q^{-6}-13q^{-8}+23q^{-10}-26q^{-12}+21q^{-14}-13q^{-16}-5q^{-18}+20q^{-20}-31q^{-22}+33q^{-24}-20q^{-26}+2q^{-28}+14q^{-30}-23q^{-32}+15q^{-34}-2q^{-36}-13q^{-38}+24q^{-40}-23q^{-42}+9q^{-44}+17q^{-46}-34q^{-48}+46q^{-50}-42q^{-52}+21q^{-54}+6q^{-56}-28q^{-58}+43q^{-60}-44q^{-62}+36q^{-64}-11q^{-66}-11q^{-68}+26q^{-70}-30q^{-72}+20q^{-74}-q^{-76}-15q^{-78}+21q^{-80}-15q^{-82}+3q^{-84}+18q^{-86}-31q^{-88}+33q^{-90}-23q^{-92}+18q^{-96}-32q^{-98}+33q^{-100}-24q^{-102}+10q^{-104}+3q^{-106}-15q^{-108}+17q^{-110}-15q^{-112}+9q^{-114}-3q^{-116}-2q^{-118}+3q^{-120}-4q^{-122}+3q^{-124}-q^{-126}+q^{-128}$

.

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 15"];

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 -2 t^2+10 t-15+10 t^{-1} -2 t^{-2} }

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 39, 2 }

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^8+2 q^7-4 q^6+6 q^5-6 q^4+7 q^3-6 q^2+4 q-2+ q^{-1} }

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^{-2} -z^4 a^{-4} -z^2 a^{-2} +2 z^2 a^{-6} +z^2- a^{-2} + a^{-4} + a^{-6} - a^{-8} +1}

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^{-4} +z^8 a^{-6} +2 z^7 a^{-3} +4 z^7 a^{-5} +2 z^7 a^{-7} +2 z^6 a^{-2} +z^6 a^{-4} +z^6 a^{-6} +2 z^6 a^{-8} +2 z^5 a^{-1} -z^5 a^{-3} -7 z^5 a^{-5} -3 z^5 a^{-7} +z^5 a^{-9} -4 z^4 a^{-6} -5 z^4 a^{-8} +z^4-3 z^3 a^{-1} +5 z^3 a^{-5} -z^3 a^{-7} -3 z^3 a^{-9} -3 z^2 a^{-2} -2 z^2 a^{-4} +2 z^2 a^{-6} +3 z^2 a^{-8} -2 z^2+z a^{-1} +z a^{-3} -z a^{-5} +z a^{-7} +2 z a^{-9} + a^{-2} + a^{-4} - a^{-6} - a^{-8} +1}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_165, K11n63, K11n101,}

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 15"];

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]=

{ Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle -2t^{2}+10t-15+10t^{-1}-2t^{-2}} , $-q^{8}+2q^{7}-4q^{6}+6q^{5}-6q^{4}+7q^{3}-6q^{2}+4q-2+q^{-1}$ }

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]=

{10_165, K11n63, K11n101,}

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:

(2, 5)

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 $8$ $40$ $32$ ${\frac {508}{3}}$ ${\frac {92}{3}}$ $320$ ${\frac {2416}{3}}$ ${\frac {352}{3}}$ $168$ ${\frac {256}{3}}$ $800$ ${\frac {4064}{3}}$ ${\frac {736}{3}}$ ${\frac {59071}{15}}$ $-{\frac {1628}{5}}$ ${\frac {92164}{45}}$ ${\frac {881}{9}}$ ${\frac {4351}{15}}$

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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (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 $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$2 is the signature of 9 15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ 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=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 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 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}} 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}\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=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}^{3}\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}^{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=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}^{4}\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=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}^{3}\oplus{\mathbb Z}_2^{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}^{4}} $r=3$ 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^{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=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}\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}\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} 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^{23}-2 q^{22}+6 q^{20}-8 q^{19}-4 q^{18}+19 q^{17}-14 q^{16}-15 q^{15}+35 q^{14}-15 q^{13}-28 q^{12}+45 q^{11}-12 q^{10}-36 q^9+45 q^8-7 q^7-33 q^6+33 q^5-q^4-21 q^3+16 q^2+q-8+5 q^{-1} -2 q^{-3} + 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^{45}+2 q^{44}-2 q^{42}-3 q^{41}+7 q^{40}+5 q^{39}-10 q^{38}-14 q^{37}+16 q^{36}+24 q^{35}-14 q^{34}-44 q^{33}+13 q^{32}+60 q^{31}-q^{30}-79 q^{29}-17 q^{28}+97 q^{27}+35 q^{26}-105 q^{25}-60 q^{24}+116 q^{23}+76 q^{22}-113 q^{21}-100 q^{20}+118 q^{19}+106 q^{18}-107 q^{17}-119 q^{16}+101 q^{15}+116 q^{14}-82 q^{13}-114 q^{12}+66 q^{11}+102 q^{10}-46 q^9-84 q^8+28 q^7+64 q^6-13 q^5-46 q^4+8 q^3+26 q^2-2 q-16+3 q^{-1} +7 q^{-2} - q^{-3} -5 q^{-4} +3 q^{-5} + q^{-6} -2 q^{-8} + q^{-9} }
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^{74}-2 q^{73}+2 q^{71}-q^{70}+4 q^{69}-9 q^{68}-q^{67}+10 q^{66}+14 q^{64}-30 q^{63}-15 q^{62}+22 q^{61}+13 q^{60}+54 q^{59}-58 q^{58}-59 q^{57}+3 q^{56}+23 q^{55}+152 q^{54}-49 q^{53}-112 q^{52}-78 q^{51}-26 q^{50}+280 q^{49}+35 q^{48}-110 q^{47}-199 q^{46}-167 q^{45}+374 q^{44}+169 q^{43}-25 q^{42}-296 q^{41}-358 q^{40}+392 q^{39}+292 q^{38}+113 q^{37}-343 q^{36}-535 q^{35}+358 q^{34}+372 q^{33}+243 q^{32}-347 q^{31}-651 q^{30}+298 q^{29}+401 q^{28}+339 q^{27}-311 q^{26}-694 q^{25}+215 q^{24}+373 q^{23}+392 q^{22}-224 q^{21}-649 q^{20}+106 q^{19}+278 q^{18}+386 q^{17}-101 q^{16}-507 q^{15}+12 q^{14}+135 q^{13}+302 q^{12}+9 q^{11}-307 q^{10}-26 q^9+15 q^8+174 q^7+49 q^6-138 q^5-8 q^4-31 q^3+66 q^2+33 q-47+13 q^{-1} -24 q^{-2} +16 q^{-3} +10 q^{-4} -17 q^{-5} +14 q^{-6} -8 q^{-7} +3 q^{-8} + q^{-9} -7 q^{-10} +6 q^{-11} - q^{-12} + q^{-13} -2 q^{-15} + q^{-16} }
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^{110}+2 q^{109}-2 q^{107}+q^{106}-2 q^{104}+5 q^{103}+2 q^{102}-9 q^{101}-3 q^{100}+3 q^{99}+2 q^{98}+16 q^{97}+9 q^{96}-20 q^{95}-29 q^{94}-12 q^{93}+11 q^{92}+50 q^{91}+54 q^{90}-11 q^{89}-71 q^{88}-92 q^{87}-40 q^{86}+80 q^{85}+160 q^{84}+104 q^{83}-44 q^{82}-203 q^{81}-234 q^{80}-35 q^{79}+234 q^{78}+343 q^{77}+197 q^{76}-177 q^{75}-483 q^{74}-406 q^{73}+65 q^{72}+552 q^{71}+644 q^{70}+162 q^{69}-568 q^{68}-898 q^{67}-440 q^{66}+505 q^{65}+1102 q^{64}+761 q^{63}-335 q^{62}-1276 q^{61}-1109 q^{60}+146 q^{59}+1372 q^{58}+1410 q^{57}+124 q^{56}-1421 q^{55}-1719 q^{54}-342 q^{53}+1418 q^{52}+1924 q^{51}+616 q^{50}-1401 q^{49}-2136 q^{48}-787 q^{47}+1341 q^{46}+2242 q^{45}+1010 q^{44}-1285 q^{43}-2365 q^{42}-1124 q^{41}+1188 q^{40}+2378 q^{39}+1299 q^{38}-1078 q^{37}-2400 q^{36}-1382 q^{35}+916 q^{34}+2309 q^{33}+1499 q^{32}-720 q^{31}-2188 q^{30}-1539 q^{29}+489 q^{28}+1958 q^{27}+1549 q^{26}-241 q^{25}-1669 q^{24}-1482 q^{23}+q^{22}+1332 q^{21}+1338 q^{20}+193 q^{19}-970 q^{18}-1129 q^{17}-328 q^{16}+630 q^{15}+900 q^{14}+368 q^{13}-357 q^{12}-632 q^{11}-356 q^{10}+144 q^9+423 q^8+291 q^7-39 q^6-228 q^5-209 q^4-32 q^3+120 q^2+128 q+39-38 q^{-1} -71 q^{-2} -41 q^{-3} +17 q^{-4} +26 q^{-5} +19 q^{-6} +13 q^{-7} -13 q^{-8} -17 q^{-9} +2 q^{-10} -2 q^{-11} -2 q^{-12} +12 q^{-13} +2 q^{-14} -6 q^{-15} +3 q^{-16} -3 q^{-17} -5 q^{-18} +4 q^{-19} +2 q^{-20} - q^{-21} + q^{-22} -2 q^{-24} + q^{-25} }
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 q^{153}-2 q^{152}+2 q^{150}-q^{149}-2 q^{147}+6 q^{146}-6 q^{145}-3 q^{144}+11 q^{143}-q^{142}-2 q^{141}-13 q^{140}+12 q^{139}-14 q^{138}-8 q^{137}+36 q^{136}+14 q^{135}+4 q^{134}-42 q^{133}+9 q^{132}-56 q^{131}-40 q^{130}+78 q^{129}+73 q^{128}+74 q^{127}-47 q^{126}+18 q^{125}-169 q^{124}-189 q^{123}+32 q^{122}+127 q^{121}+245 q^{120}+106 q^{119}+225 q^{118}-239 q^{117}-465 q^{116}-296 q^{115}-103 q^{114}+287 q^{113}+379 q^{112}+870 q^{111}+139 q^{110}-498 q^{109}-797 q^{108}-869 q^{107}-338 q^{106}+251 q^{105}+1722 q^{104}+1220 q^{103}+356 q^{102}-797 q^{101}-1808 q^{100}-1854 q^{99}-973 q^{98}+1961 q^{97}+2517 q^{96}+2241 q^{95}+419 q^{94}-2012 q^{93}-3651 q^{92}-3328 q^{91}+884 q^{90}+3095 q^{89}+4451 q^{88}+2782 q^{87}-882 q^{86}-4793 q^{85}-6017 q^{84}-1336 q^{83}+2457 q^{82}+6087 q^{81}+5471 q^{80}+1273 q^{79}-4876 q^{78}-8170 q^{77}-3863 q^{76}+959 q^{75}+6803 q^{74}+7686 q^{73}+3621 q^{72}-4237 q^{71}-9459 q^{70}-5964 q^{69}-679 q^{68}+6860 q^{67}+9134 q^{66}+5537 q^{65}-3414 q^{64}-10060 q^{63}-7406 q^{62}-2014 q^{61}+6618 q^{60}+9940 q^{59}+6889 q^{58}-2632 q^{57}-10192 q^{56}-8327 q^{55}-3070 q^{54}+6131 q^{53}+10254 q^{52}+7867 q^{51}-1723 q^{50}-9805 q^{49}-8855 q^{48}-4092 q^{47}+5144 q^{46}+9955 q^{45}+8556 q^{44}-408 q^{43}-8580 q^{42}-8795 q^{41}-5116 q^{40}+3405 q^{39}+8681 q^{38}+8653 q^{37}+1237 q^{36}-6307 q^{35}-7713 q^{34}-5715 q^{33}+1143 q^{32}+6283 q^{31}+7664 q^{30}+2591 q^{29}-3415 q^{28}-5504 q^{27}-5290 q^{26}-813 q^{25}+3352 q^{24}+5548 q^{23}+2923 q^{22}-933 q^{21}-2863 q^{20}-3801 q^{19}-1647 q^{18}+976 q^{17}+3078 q^{16}+2182 q^{15}+326 q^{14}-831 q^{13}-2006 q^{12}-1377 q^{11}-174 q^{10}+1227 q^9+1096 q^8+480 q^7+99 q^6-724 q^5-722 q^4-356 q^3+333 q^2+346 q+231+252 q^{-1} -159 q^{-2} -254 q^{-3} -206 q^{-4} +67 q^{-5} +50 q^{-6} +41 q^{-7} +159 q^{-8} -11 q^{-9} -62 q^{-10} -79 q^{-11} +18 q^{-12} -10 q^{-13} -17 q^{-14} +69 q^{-15} +7 q^{-16} -8 q^{-17} -25 q^{-18} +10 q^{-19} -10 q^{-20} -18 q^{-21} +24 q^{-22} +3 q^{-23} +3 q^{-24} -7 q^{-25} +5 q^{-26} -3 q^{-27} -9 q^{-28} +6 q^{-29} +2 q^{-31} - q^{-32} + q^{-33} -2 q^{-35} + q^{-36} }

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.

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