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DrawMorseLink

PD Presentation:

X6172 X16,9,17,10 X4,21,1,22 X14,12,15,11 X10,4,11,3 X12,5,13,6 X22,13,5,14 X2,16,3,15 X20,18,21,17 X18,8,19,7 X8,20,9,19

Gauss Code:

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

Jones Polynomial:

q-9/2 - 4q-7/2 + 9q-5/2 - 16q-3/2 + 21q-1/2 - 26q1/2 + 26q3/2 - 23q5/2 + 17q7/2 - 11q9/2 + 5q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-14 + q-12 + q-10 - 3q-8 + 5q-6 - q-4 + 6 - 3q2 + 5q4 - 3q6 + q8 + 3q10 - 4q12 + 4q14 - q16 - 2q18 + q20

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-7z5 + 4a-6z4 - 5a-6z6 - 5a-5z3 + 15a-5z5 - 11a-5z7 - 5a-4z4 + 17a-4z6 - 13a-4z8 + 2a-3z - 16a-3z3 + 28a-3z5 - 5a-3z7 - 8a-3z9 + 3a-2z2 - 29a-2z4 + 51a-2z6 - 23a-2z8 - 2a-2z10 - a-1z-1 + 6a-1z - 22a-1z3 + 20a-1z5 + 12a-1z7 - 14a-1z9 + 1 + 5z2 - 30z4 + 44z6 - 17z8 - 2z10 - az-1 + 6az - 18az3 + 17az5 + 2az7 - 6az9 + a2z2 - 8a2z4 + 14a2z6 - 7a2z8 + 2a3z - 7a3z3 + 9a3z5 - 4a3z7 - a4z2 + 2a4z4 - 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 10         j = 2           13 13           j = 0         10 15             j = -2       6 11               j = -4     3 10                 j = -6   1 6                   j = -8   3                     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, 49]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 49]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 9, 17, 10], X[4, 21, 1, 22], X[14, 12, 15, 11], > X[10, 4, 11, 3], X[12, 5, 13, 6], X[22, 13, 5, 14], X[2, 16, 3, 15], > X[20, 18, 21, 17], X[18, 8, 19, 7], X[8, 20, 9, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 5, -3}, {6, -1, 10, -11, 2, -5, 4, -6, 7, -4, 8, -2, 9, -10, > 11, -9, 3, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 4 9 16 21 3/2 5/2 q - ---- + ---- - ---- + ------- - 26 Sqrt[q] + 26 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 -12 -10 3 5 -4 2 4 6 8 10 6 - q + q + q - -- + -- - q - 3 q + 5 q - 3 q + q + 3 q - 8 6 q q 12 14 16 18 20 > 4 q + 4 q - q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 49]][a, z]
Out[8]=	3 3 3 1 a z 3 z 3 z 3 z 5 z 3 3 3 -(---) + - + -- - --- + 3 a z - a z - -- + ---- - ---- + 4 a z - a z + a z z 3 a 5 3 a a a a 5 5 7 2 z 3 z 5 z > ---- - ---- + 2 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 49]][a, z]
Out[9]=	2 3 1 a 2 z 6 z 3 2 3 z 2 2 4 2 5 z 1 - --- - - + --- + --- + 6 a z + 2 a z + 5 z + ---- + a z - a z - ---- - a z z 3 a 2 5 a a a 3 3 4 4 4 16 z 22 z 3 3 3 4 4 z 5 z 29 z 2 4 > ----- - ----- - 18 a z - 7 a z - 30 z + ---- - ---- - ----- - 8 a z + 3 a 6 4 2 a a a a 5 5 5 5 6 4 4 z 15 z 28 z 20 z 5 3 5 6 5 z > 2 a z - -- + ----- + ----- + ----- + 17 a z + 9 a z + 44 z - ---- + 7 5 3 a 6 a a a a 6 6 7 7 7 17 z 51 z 2 6 4 6 11 z 5 z 12 z 7 > ----- + ----- + 14 a z - a z - ----- - ---- + ----- + 2 a z - 4 2 5 3 a a a a a 8 8 9 9 3 7 8 13 z 23 z 2 8 8 z 14 z 9 10 > 4 a z - 17 z - ----- - ----- - 7 a z - ---- - ----- - 6 a z - 2 z - 4 2 3 a a a a 10 2 z > ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 6 3 10 6 10 15 + 13 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 > ---- + 13 q t + 13 q t + 10 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 L11a49

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