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Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,3,13,4 X18,8,19,7 X16,6,17,5 X22,13,9,14 X20,15,21,16 X6,18,7,17 X14,21,15,22 X4,20,5,19 X2,9,3,10 X8,11,1,12

Gauss Code:

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

Jones Polynomial:

q-13/2 - 2q-11/2 + 4q-9/2 - 7q-7/2 + 8q-5/2 - 11q-3/2 + 11q-1/2 - 10q1/2 + 8q3/2 - 6q5/2 + 3q7/2 - q9/2

A2 (sl(3)) Invariant:

- q-20 - q-14 + 2q-12 + q-8 + 3q-6 + 2q-2 + 2q4 - q6 + 2q8 - q12 + q14

HOMFLY-PT Polynomial:

- a-3z - a-3z3 - a-1z-1 - 2a-1z + a-1z3 + a-1z5 + az-1 + 5az + 6az3 + 2az5 + 2a3z3 + a3z5 - 2a5z - a5z3

Kauffman Polynomial:

- a-5z3 - 3a-4z4 - 2a-3z + 5a-3z3 - 6a-3z5 - 4a-2z2 + 11a-2z4 - 8a-2z6 - a-1z-1 + 4a-1z - 6a-1z3 + 14a-1z5 - 8a-1z7 + 1 - z2 + 2z4 + 10z6 - 6z8 - az-1 + 10az - 26az3 + 23az5 - 3az9 + 4a2z2 - 22a2z4 + 23a2z6 - 4a2z8 - a2z10 - 17a3z5 + 19a3z7 - 5a3z9 + 9a4z2 - 22a4z4 + 11a4z6 + a4z8 - a4z10 - 4a5z + 14a5z3 - 20a5z5 + 11a5z7 - 2a5z9 + 8a6z2 - 12a6z4 + 6a6z6 - a6z8

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 10                       1 j = 8                     2   j = 6                   4 1   j = 4                 4 2     j = 2               6 4       j = 0             7 6         j = -2           4 4           j = -4         4 7             j = -6       3 4               j = -8     1 4                 j = -10   1 3                   j = -12   1                     j = -14 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 298]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 298]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[18, 8, 19, 7], X[16, 6, 17, 5], > X[22, 13, 9, 14], X[20, 15, 21, 16], X[6, 18, 7, 17], X[14, 21, 15, 22], > X[4, 20, 5, 19], X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -9, 4, -7, 3, -11}, > {10, -1, 11, -2, 5, -8, 6, -4, 7, -3, 9, -6, 8, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 4 7 8 11 11 3/2 q - ----- + ---- - ---- + ---- - ---- + ------- - 10 Sqrt[q] + 8 q - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 7/2 9/2 > 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -14 2 -8 3 2 4 6 8 12 14 -q - q + --- + q + -- + -- + 2 q - q + 2 q - q + q 12 6 2 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 298]][a, z]
Out[8]=	3 3 1 a z 2 z 5 z z 3 3 3 5 3 -(---) + - - -- - --- + 5 a z - 2 a z - -- + -- + 6 a z + 2 a z - a z + a z z 3 a 3 a a a 5 z 5 3 5 > -- + 2 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 298]][a, z]
Out[9]=	2 1 a 2 z 4 z 5 2 4 z 2 2 4 2 1 - --- - - - --- + --- + 10 a z - 4 a z - z - ---- + 4 a z + 9 a z + a z z 3 a 2 a a 3 3 3 4 4 6 2 z 5 z 6 z 3 5 3 4 3 z 11 z > 8 a z - -- + ---- - ---- - 26 a z + 14 a z + 2 z - ---- + ----- - 5 3 a 4 2 a a a a 5 5 2 4 4 4 6 4 6 z 14 z 5 3 5 > 22 a z - 22 a z - 12 a z - ---- + ----- + 23 a z - 17 a z - 3 a a 6 7 5 5 6 8 z 2 6 4 6 6 6 8 z 3 7 > 20 a z + 10 z - ---- + 23 a z + 11 a z + 6 a z - ---- + 19 a z + 2 a a 5 7 8 2 8 4 8 6 8 9 3 9 5 9 > 11 a z - 6 z - 4 a z + a z - a z - 3 a z - 5 a z - 2 a z - 2 10 4 10 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 3 1 4 3 4 6 + 6 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 14 7 12 6 10 6 10 5 8 5 8 4 6 4 6 3 q t q t q t q t q t q t q t q t 4 7 4 7 4 2 4 4 2 6 2 > ----- + ----- + ----- + - + ---- + 4 q t + 4 q t + 2 q t + 4 q t + 4 3 4 2 2 2 t 2 q t q t q t q t 6 3 8 3 10 4 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a298

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