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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9/2 - 3q-7/2 + 8q-5/2 - 15q-3/2 + 20q-1/2 - 25q1/2 + 25q3/2 - 23q5/2 + 17q7/2 - 11q9/2 + 5q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-14 - 4q-8 + 4q-6 - q-4 + 7 - 2q2 + 6q4 - 2q6 + q8 + 3q10 - 4q12 + 4q14 - q16 - 2q18 + q20

HOMFLY-PT Polynomial:

- a-5z3 + a-3z + 3a-3z3 + 2a-3z5 - 2a-1z-1 - 4a-1z - 5a-1z3 - 3a-1z5 - a-1z7 + 3az-1 + 6az + 5az3 + 2az5 - a3z-1 - 2a3z - a3z3

Kauffman Polynomial:

- a-7z5 + 4a-6z4 - 5a-6z6 - 5a-5z3 + 15a-5z5 - 11a-5z7 - 4a-4z4 + 17a-4z6 - 13a-4z8 + 2a-3z - 16a-3z3 + 27a-3z5 - 4a-3z7 - 8a-3z9 + 2a-2z2 - 25a-2z4 + 46a-2z6 - 21a-2z8 - 2a-2z10 - 2a-1z-1 + 9a-1z - 21a-1z3 + 16a-1z5 + 12a-1z7 - 13a-1z9 + 3 - 21z4 + 33z6 - 13z8 - 2z10 - 3az-1 + 10az - 16az3 + 12az5 + 2az7 - 5az9 + 3a2 - 5a2z2 - a2z4 + 8a2z6 - 5a2z8 - a3z-1 + 3a3z - 6a3z3 + 7a3z5 - 3a3z7 + a4 - 3a4z2 + 3a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 14                       1 j = 12                     4   j = 10                   7 1   j = 8                 10 4     j = 6               13 7       j = 4             13 11         j = 2           12 12           j = 0         9 14             j = -2       6 11               j = -4     2 9                 j = -6   1 6                   j = -8   2                     j = -10 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, 237]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 237]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 4, 13, 3], X[14, 12, 15, 11], X[22, 15, 7, 16], > X[16, 9, 17, 10], X[10, 21, 11, 22], X[18, 6, 19, 5], X[20, 18, 21, 17], > X[2, 7, 3, 8], X[4, 14, 5, 13], X[6, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 7, -11}, > {9, -1, 5, -6, 3, -2, 10, -3, 4, -5, 8, -7, 11, -8, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 8 15 20 3/2 5/2 q - ---- + ---- - ---- + ------- - 25 Sqrt[q] + 25 q - 23 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 17 q - 11 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 4 4 -4 2 4 6 8 10 12 14 7 - q - -- + -- - q - 2 q + 6 q - 2 q + q + 3 q - 4 q + 4 q - 8 6 q q 16 18 20 > q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 237]][a, z]
Out[8]=	3 3 3 3 -2 3 a a z 4 z 3 z 3 z 5 z 3 --- + --- - -- + -- - --- + 6 a z - 2 a z - -- + ---- - ---- + 5 a z - a z z z 3 a 5 3 a a a a 5 5 7 3 3 2 z 3 z 5 z > a z + ---- - ---- + 2 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 237]][a, z]
Out[9]=	3 2 2 4 2 3 a a 2 z 9 z 3 2 z 2 2 3 + 3 a + a - --- - --- - -- + --- + --- + 10 a z + 3 a z + ---- - 5 a z - a z z z 3 a 2 a a 3 3 3 4 4 4 2 5 z 16 z 21 z 3 3 3 4 4 z 4 z > 3 a z - ---- - ----- - ----- - 16 a z - 6 a z - 21 z + ---- - ---- - 5 3 a 6 4 a a a a 4 5 5 5 5 25 z 2 4 4 4 z 15 z 27 z 16 z 5 3 5 > ----- - a z + 3 a z - -- + ----- + ----- + ----- + 12 a z + 7 a z + 2 7 5 3 a a a a a 6 6 6 7 7 7 6 5 z 17 z 46 z 2 6 4 6 11 z 4 z 12 z > 33 z - ---- + ----- + ----- + 8 a z - a z - ----- - ---- + ----- + 6 4 2 5 3 a a a a a a 8 8 9 9 7 3 7 8 13 z 21 z 2 8 8 z 13 z > 2 a z - 3 a z - 13 z - ----- - ----- - 5 a z - ---- - ----- - 4 2 3 a a a a 10 9 10 2 z > 5 a z - 2 z - ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 6 2 9 6 9 14 + 12 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t q t q t q t q t q t q t q t 11 2 4 4 2 6 2 6 3 8 3 > ---- + 12 q t + 13 q t + 11 q t + 13 q t + 7 q t + 10 q t + 2 q t 8 4 10 4 10 5 12 5 14 6 > 4 q t + 7 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a237

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