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DrawMorseLink

PD Presentation:

X12,1,13,2 X8493 X16,6,17,5 X22,8,11,7 X20,15,21,16 X14,21,15,22 X6,14,7,13 X4,20,5,19 X18,9,19,10 X2,11,3,12 X10,17,1,18

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 4q-9/2 - 9q-7/2 + 15q-5/2 - 21q-3/2 + 24q-1/2 - 25q1/2 + 21q3/2 - 17q5/2 + 10q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

q-18 - 3q-14 + 2q-12 - 2q-8 + 6q-6 - 2q-4 + 2q-2 + 1 - 2q2 + 6q4 - 3q6 + 5q8 + q10 - 4q12 + 2q14 - q16

HOMFLY-PT Polynomial:

3a-3z + 2a-3z3 + a-3z5 - a-1z-1 - 7a-1z - 9a-1z3 - 4a-1z5 - a-1z7 + az-1 + 6az + 7az3 + 3az5 - 3a3z - 3a3z3 + a5z

Kauffman Polynomial:

- a-6z4 - 4a-5z5 - 3a-4z2 + 7a-4z4 - 10a-4z6 + 5a-3z - 16a-3z3 + 25a-3z5 - 17a-3z7 + a-2z2 - 12a-2z4 + 29a-2z6 - 18a-2z8 - a-1z-1 + 10a-1z - 29a-1z3 + 32a-1z5 + 4a-1z7 - 11a-1z9 + 1 + 10z2 - 49z4 + 68z6 - 21z8 - 3z10 - az-1 + 6az - 14az3 - 9az5 + 38az7 - 17az9 + 11a2z2 - 43a2z4 + 42a2z6 - 7a2z8 - 3a2z10 + 2a3z - 4a3z3 - 9a3z5 + 16a3z7 - 6a3z9 + 5a4z2 - 14a4z4 + 13a4z6 - 4a4z8 + a5z - 3a5z3 + 3a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                       1 j = 10                     3   j = 8                   7 1   j = 6                 10 3     j = 4               11 7       j = 2             14 10         j = 0           12 13           j = -2         9 12             j = -4       6 12               j = -6     3 9                 j = -8   1 6                   j = -10   3                     j = -12 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, 353]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 353]]
Out[4]=	PD[X[12, 1, 13, 2], X[8, 4, 9, 3], X[16, 6, 17, 5], X[22, 8, 11, 7], > X[20, 15, 21, 16], X[14, 21, 15, 22], X[6, 14, 7, 13], X[4, 20, 5, 19], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[10, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -8, 3, -7, 4, -2, 9, -11}, > {10, -1, 7, -6, 5, -3, 11, -9, 8, -5, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 4 9 15 21 24 3/2 -q + ---- - ---- + ---- - ---- + ------- - 25 Sqrt[q] + 21 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 17 q + 10 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 3 2 2 6 2 2 2 4 6 8 10 1 + q - --- + --- - -- + -- - -- + -- - 2 q + 6 q - 3 q + 5 q + q - 14 12 8 6 4 2 q q q q q q 12 14 16 > 4 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 353]][a, z]
Out[8]=	3 3 1 a 3 z 7 z 3 5 2 z 9 z 3 -(---) + - + --- - --- + 6 a z - 3 a z + a z + ---- - ---- + 7 a z - a z z 3 a 3 a a a 5 5 7 3 3 z 4 z 5 z > 3 a z + -- - ---- + 3 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 353]][a, z]
Out[9]=	2 2 1 a 5 z 10 z 3 5 2 3 z z 1 - --- - - + --- + ---- + 6 a z + 2 a z + a z + 10 z - ---- + -- + a z z 3 a 4 2 a a a 3 3 2 2 4 2 16 z 29 z 3 3 3 5 3 4 > 11 a z + 5 a z - ----- - ----- - 14 a z - 4 a z - 3 a z - 49 z - 3 a a 4 4 4 5 5 5 z 7 z 12 z 2 4 4 4 4 z 25 z 32 z 5 > -- + ---- - ----- - 43 a z - 14 a z - ---- + ----- + ----- - 9 a z - 6 4 2 5 3 a a a a a a 6 6 7 3 5 5 5 6 10 z 29 z 2 6 4 6 17 z > 9 a z + 3 a z + 68 z - ----- + ----- + 42 a z + 13 a z - ----- + 4 2 3 a a a 7 8 4 z 7 3 7 5 7 8 18 z 2 8 4 8 > ---- + 38 a z + 16 a z - a z - 21 z - ----- - 7 a z - 4 a z - a 2 a 9 11 z 9 3 9 10 2 10 > ----- - 17 a z - 6 a z - 3 z - 3 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 6 3 9 6 12 13 + 14 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 9 12 12 2 4 4 2 6 2 6 3 > ----- + -- + ---- + 10 q t + 11 q t + 7 q t + 10 q t + 3 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a353

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