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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X22,10,7,9 X2738 X4,22,5,21 X14,5,15,6 X16,11,17,12 X20,17,21,18 X18,14,19,13 X12,20,13,19 X6,15,1,16

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 4q-11/2 - 9q-9/2 + 14q-7/2 - 20q-5/2 + 22q-3/2 - 22q-1/2 + 19q1/2 - 14q3/2 + 8q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

q-20 - q-18 + 4q-14 - 3q-12 + 3q-10 + 2q-8 - 2q-6 + 4q-4 - 4q-2 + 3 - q2 - q4 + 5q6 - 2q8 + q10 + q12 - q14

HOMFLY-PT Polynomial:

a-3z + a-3z3 - a-1z-1 - 5a-1z - 5a-1z3 - 2a-1z5 + 2az-1 + 8az + 9az3 + 4az5 + az7 - 2a3z-1 - 6a3z - 5a3z3 - 2a3z5 + a5z-1 + a5z + a5z3

Kauffman Polynomial:

- a-4z2 + 2a-4z4 - a-4z6 + 2a-3z - 8a-3z3 + 10a-3z5 - 4a-3z7 + a-2z2 - 6a-2z4 + 12a-2z6 - 6a-2z8 - a-1z-1 + 10a-1z - 30a-1z3 + 31a-1z5 - 5a-1z7 - 4a-1z9 + 9z2 - 32z4 + 43z6 - 17z8 - z10 - 2az-1 + 18az - 48az3 + 43az5 - 2az7 - 9az9 - a2 + 10a2z2 - 35a2z4 + 46a2z6 - 20a2z8 - a2z10 - 2a3z-1 + 14a3z - 36a3z3 + 36a3z5 - 9a3z7 - 5a3z9 + 2a4z2 - 6a4z4 + 12a4z6 - 9a4z8 - a5z-1 + 4a5z - 9a5z3 + 13a5z5 - 8a5z7 - a6z2 + 5a6z4 - 4a6z6 + a7z3 - a7z5

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 = 10                       1 j = 8                     3   j = 6                   5 1   j = 4                 9 3     j = 2               10 5       j = 0             12 9         j = -2           11 11           j = -4         9 11             j = -6       6 12               j = -8     3 8                 j = -10   1 6                   j = -12   3                     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, 159]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 159]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[22, 10, 7, 9], X[2, 7, 3, 8], > X[4, 22, 5, 21], X[14, 5, 15, 6], X[16, 11, 17, 12], X[20, 17, 21, 18], > X[18, 14, 19, 13], X[12, 20, 13, 19], X[6, 15, 1, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -5, 6, -11}, > {4, -1, 3, -2, 7, -10, 9, -6, 11, -7, 8, -9, 10, -8, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 4 9 14 20 22 22 -q + ----- - ---- + ---- - ---- + ---- - ------- + 19 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 9/2 > 14 q + 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 4 3 3 2 2 4 4 2 4 6 8 3 + q - q + --- - --- + --- + -- - -- + -- - -- - q - q + 5 q - 2 q + 14 12 10 8 6 4 2 q q q q q q q 10 12 14 > q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 159]][a, z]
Out[8]=	3 5 3 3 1 2 a 2 a a z 5 z 3 5 z 5 z -(---) + --- - ---- + -- + -- - --- + 8 a z - 6 a z + a z + -- - ---- + a z z z z 3 a 3 a a a 5 3 3 3 5 3 2 z 5 3 5 7 > 9 a z - 5 a z + a z - ---- + 4 a z - 2 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 159]][a, z]
Out[9]=	3 5 2 1 2 a 2 a a 2 z 10 z 3 5 2 -a - --- - --- - ---- - -- + --- + ---- + 18 a z + 14 a z + 4 a z + 9 z - a z z z z 3 a a 2 2 3 3 z z 2 2 4 2 6 2 8 z 30 z 3 3 3 > -- + -- + 10 a z + 2 a z - a z - ---- - ----- - 48 a z - 36 a z - 4 2 3 a a a a 4 4 5 3 7 3 4 2 z 6 z 2 4 4 4 6 4 > 9 a z + a z - 32 z + ---- - ---- - 35 a z - 6 a z + 5 a z + 4 2 a a 5 5 6 10 z 31 z 5 3 5 5 5 7 5 6 z > ----- + ----- + 43 a z + 36 a z + 13 a z - a z + 43 z - -- + 3 a 4 a a 6 7 7 12 z 2 6 4 6 6 6 4 z 5 z 7 3 7 > ----- + 46 a z + 12 a z - 4 a z - ---- - ---- - 2 a z - 9 a z - 2 3 a a a 8 9 5 7 8 6 z 2 8 4 8 4 z 9 3 9 > 8 a z - 17 z - ---- - 20 a z - 9 a z - ---- - 9 a z - 5 a z - 2 a a 10 2 10 > z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	11 1 3 1 6 3 8 6 12 12 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 9 11 11 2 2 2 4 2 4 3 > ----- + ---- + ---- + 9 t + 10 q t + 5 q t + 9 q t + 3 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a159

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