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DrawMorseLink

PD Presentation:

X8192 X20,9,21,10 X14,5,15,6 X16,8,17,7 X10,4,11,3 X22,14,7,13 X18,12,19,11 X12,18,13,17 X6,15,1,16 X4,21,5,22 X2,20,3,19

Gauss Code:

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

Jones Polynomial:

q-11/2 - 4q-9/2 + 9q-7/2 - 16q-5/2 + 21q-3/2 - 25q-1/2 + 24q1/2 - 22q3/2 + 16q5/2 - 10q7/2 + 5q9/2 - q11/2

A2 (sl(3)) Invariant:

- q-16 + 2q-14 - 2q-12 + 4q-8 - 3q-6 + 6q-4 + 2 + 4q2 - 3q4 + 5q6 - 3q8 - q10 + q12 - 3q14 + q16

HOMFLY-PT Polynomial:

a-3z-1 + 2a-3z - a-3z3 - a-3z5 - 3a-1z-1 - 6a-1z - 2a-1z3 + 2a-1z5 + a-1z7 + 2az-1 + 4az + 4az3 + 3az5 + az7 - a3z - 2a3z3 - a3z5

Kauffman Polynomial:

- a-5z3 + 2a-5z5 - a-5z7 + a-4 - 12a-4z4 + 15a-4z6 - 5a-4z8 - a-3z-1 + 3a-3z + a-3z3 - 19a-3z5 + 24a-3z7 - 8a-3z9 + 3a-2 + 3a-2z2 - 36a-2z4 + 39a-2z6 - 4a-2z8 - 4a-2z10 - 3a-1z-1 + 10a-1z - 7a-1z3 - 28a-1z5 + 52a-1z7 - 20a-1z9 + 3 + 6z2 - 46z4 + 62z6 - 16z8 - 4z10 - 2az-1 + 9az - 22az3 + 18az5 + 12az7 - 12az9 + 2a2z2 - 15a2z4 + 29a2z6 - 17a2z8 + 2a3z - 12a3z3 + 21a3z5 - 15a3z7 - a4z2 + 6a4z4 - 9a4z6 + a5z3 - 4a5z5 - a6z4

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 = 12                       1 j = 10                     4   j = 8                   6 1   j = 6                 10 4     j = 4               12 6       j = 2             12 10         j = 0           13 12           j = -2         9 13             j = -4       7 12               j = -6     3 10                 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, 181]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 181]]
Out[4]=	PD[X[8, 1, 9, 2], X[20, 9, 21, 10], X[14, 5, 15, 6], X[16, 8, 17, 7], > X[10, 4, 11, 3], X[22, 14, 7, 13], X[18, 12, 19, 11], X[12, 18, 13, 17], > X[6, 15, 1, 16], X[4, 21, 5, 22], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -10, 3, -9}, > {4, -1, 2, -5, 7, -8, 6, -3, 9, -4, 8, -7, 11, -2, 10, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 4 9 16 21 25 3/2 q - ---- + ---- - ---- + ---- - ------- + 24 Sqrt[q] - 22 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 16 q - 10 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 2 4 3 6 2 4 6 8 10 12 2 - q + --- - --- + -- - -- + -- + 4 q - 3 q + 5 q - 3 q - q + q - 14 12 8 6 4 q q q q q 14 16 > 3 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 181]][a, z]
Out[8]=	3 3 1 3 2 a 2 z 6 z 3 z 2 z 3 3 3 ---- - --- + --- + --- - --- + 4 a z - a z - -- - ---- + 4 a z - 2 a z - 3 a z z 3 a 3 a a z a a 5 5 7 z 2 z 5 3 5 z 7 > -- + ---- + 3 a z - a z + -- + a z 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 181]][a, z]
Out[9]=	2 -4 3 1 3 2 a 3 z 10 z 3 2 3 z 3 + a + -- - ---- - --- - --- + --- + ---- + 9 a z + 2 a z + 6 z + ---- + 2 3 a z z 3 a 2 a a z a a 3 3 3 2 2 4 2 z z 7 z 3 3 3 5 3 4 > 2 a z - a z - -- + -- - ---- - 22 a z - 12 a z + a z - 46 z - 5 3 a a a 4 4 5 5 5 12 z 36 z 2 4 4 4 6 4 2 z 19 z 28 z > ----- - ----- - 15 a z + 6 a z - a z + ---- - ----- - ----- + 4 2 5 3 a a a a a 6 6 5 3 5 5 5 6 15 z 39 z 2 6 4 6 > 18 a z + 21 a z - 4 a z + 62 z + ----- + ----- + 29 a z - 9 a z - 4 2 a a 7 7 7 8 8 z 24 z 52 z 7 3 7 8 5 z 4 z 2 8 > -- + ----- + ----- + 12 a z - 15 a z - 16 z - ---- - ---- - 17 a z - 5 3 a 4 2 a a a a 9 9 10 8 z 20 z 9 10 4 z > ---- - ----- - 12 a z - 4 z - ----- 3 a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	13 1 3 1 6 3 10 7 12 13 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 9 2 2 2 4 2 4 3 6 3 > ---- + 12 t + 12 q t + 10 q t + 12 q t + 6 q t + 10 q t + 2 q t 6 4 8 4 8 5 10 5 12 6 > 4 q t + 6 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a181

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