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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 4q-11/2 - 9q-9/2 + 16q-7/2 - 21q-5/2 + 24q-3/2 - 25q-1/2 + 20q1/2 - 16q3/2 + 9q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 - 2q-16 + 3q-14 - 4q-12 - 5q-6 + 6q-4 - 3q-2 + 8 + 2q2 + q4 + 4q6 - 3q8 + 2q10 - q12

HOMFLY-PT Polynomial:

- 2a-1z-1 + a-1z + 5a-1z3 + 4a-1z5 + a-1z7 + 3az-1 - az - 11az3 - 13az5 - 6az7 - az9 - a3z-1 + a3z + 5a3z3 + 4a3z5 + a3z7

Kauffman Polynomial:

- a-4z2 + 2a-4z4 - a-4z6 + a-3z - 6a-3z3 + 9a-3z5 - 4a-3z7 - a-2z2 - 6a-2z4 + 14a-2z6 - 7a-2z8 - 2a-1z-1 + 2a-1z - 2a-1z3 + 10a-1z7 - 7a-1z9 + 3 - z2 - 12z4 + 24z6 - 8z8 - 3z10 - 3az-1 - az + 12az3 - 20az5 + 28az7 - 15az9 + 3a2 - 14a2z4 + 26a2z6 - 11a2z8 - 3a2z10 - a3z-1 - 2a3z + 2a3z3 + a3z5 + 6a3z7 - 8a3z9 + a4 - a4z2 - 5a4z4 + 13a4z6 - 10a4z8 - 5a5z3 + 11a5z5 - 8a5z7 - 2a6z2 + 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                   6 1   j = 4                 10 3     j = 2               11 7       j = 0             14 9         j = -2           11 12           j = -4         10 13             j = -6       6 11               j = -8     3 10                 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, 177]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 177]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[16, 7, 17, 8], X[18, 10, 19, 9], > X[20, 11, 21, 12], X[22, 15, 7, 16], X[12, 21, 13, 22], X[14, 6, 15, 5], > X[4, 14, 5, 13], X[6, 18, 1, 17], X[2, 19, 3, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -9, 8, -10}, > {3, -1, 4, -2, 5, -7, 9, -8, 6, -3, 10, -4, 11, -5, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 4 9 16 21 24 25 -q + ----- - ---- + ---- - ---- + ---- - ------- + 20 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 > 16 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 3 4 5 6 3 2 4 6 8 10 8 + q - --- + --- - --- - -- + -- - -- + 2 q + q + 4 q - 3 q + 2 q - 16 14 12 6 4 2 q q q q q q 12 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 177]][a, z]
Out[8]=	3 3 5 -2 3 a a z 3 5 z 3 3 3 4 z 5 --- + --- - -- + - - a z + a z + ---- - 11 a z + 5 a z + ---- - 13 a z + a z z z a a a 7 3 5 z 7 3 7 9 > 4 a z + -- - 6 a z + a z - a z a
In[9]:=	Kauffman[Link[11, Alternating, 177]][a, z]
Out[9]=	3 2 2 2 4 2 3 a a z 2 z 3 2 z z 3 + 3 a + a - --- - --- - -- + -- + --- - a z - 2 a z - z - -- - -- - a z z z 3 a 4 2 a a a 3 3 4 2 6 2 6 z 2 z 3 3 3 5 3 7 3 > a z - 2 a z - ---- - ---- + 12 a z + 2 a z - 5 a z + a z - 3 a a 4 4 5 4 2 z 6 z 2 4 4 4 6 4 9 z 5 > 12 z + ---- - ---- - 14 a z - 5 a z + 5 a z + ---- - 20 a z + 4 2 3 a a a 6 6 3 5 5 5 7 5 6 z 14 z 2 6 4 6 > a z + 11 a z - a z + 24 z - -- + ----- + 26 a z + 13 a z - 4 2 a a 7 7 8 6 6 4 z 10 z 7 3 7 5 7 8 7 z > 4 a z - ---- + ----- + 28 a z + 6 a z - 8 a z - 8 z - ---- - 3 a 2 a a 9 2 8 4 8 7 z 9 3 9 10 2 10 > 11 a z - 10 a z - ---- - 15 a z - 8 a z - 3 z - 3 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	12 1 3 1 6 3 10 6 11 14 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 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 10 13 11 2 2 2 4 2 4 3 > ----- + ---- + ---- + 9 t + 11 q t + 7 q t + 10 q t + 3 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a177

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