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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,3,13,4 X18,8,19,7 X14,6,15,5 X22,17,9,18 X16,21,17,22 X20,13,21,14 X6,16,7,15 X4,20,5,19 X2,9,3,10 X8,11,1,12

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 7q-11/2 - 12q-9/2 + 17q-7/2 - 21q-5/2 + 20q-3/2 - 19q-1/2 + 14q1/2 - 9q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-22 + q-20 - 2q-18 - q-16 + 2q-14 - 4q-12 + 4q-10 + q-8 + 2q-6 + 5q-4 - 2q-2 + 5 - 2q2 + 2q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- a-1z - 2a-1z3 - a-1z5 - 2az-1 - 2az + 2az3 + 3az5 + az7 + 3a3z-1 + 6a3z + 7a3z3 + 4a3z5 + a3z7 - a5z-1 - 3a5z - 3a5z3 - a5z5

Kauffman Polynomial:

a-3z3 - a-3z5 - a-2z2 + 5a-2z4 - 4a-2z6 + 2a-1z - 7a-1z3 + 13a-1z5 - 8a-1z7 + z2 - 5z4 + 13z6 - 9z8 + 2az-1 - 3az - 4az3 + 8az5 + 2az7 - 6az9 - 3a2 + 8a2z2 - 20a2z4 + 25a2z6 - 10a2z8 - 2a2z10 + 3a3z-1 - 11a3z + 22a3z3 - 27a3z5 + 24a3z7 - 11a3z9 - 3a4 + 14a4z2 - 25a4z4 + 22a4z6 - 6a4z8 - 2a4z10 + a5z-1 - 6a5z + 13a5z3 - 13a5z5 + 11a5z7 - 5a5z9 - a6 + 6a6z2 - 12a6z4 + 13a6z6 - 5a6z8 - 5a7z3 + 8a7z5 - 3a7z7 - 2a8z2 + 3a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                       1 j = 6                     3   j = 4                   6 1   j = 2                 8 3     j = 0               11 6       j = -2             11 10         j = -4           10 9           j = -6         7 11             j = -8       5 10               j = -10     2 7                 j = -12   1 5                   j = -14   2                     j = -16 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, 261]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 261]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[18, 8, 19, 7], X[14, 6, 15, 5], > X[22, 17, 9, 18], X[16, 21, 17, 22], X[20, 13, 21, 14], X[6, 16, 7, 15], > X[4, 20, 5, 19], X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -9, 4, -8, 3, -11}, > {10, -1, 11, -2, 7, -4, 8, -6, 5, -3, 9, -7, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 7 12 17 21 20 19 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 14 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 2 -16 2 4 4 -8 2 5 2 2 5 - q + q - --- - q + --- - --- + --- + q + -- + -- - -- - 2 q + 18 14 12 10 6 4 2 q q q q q q q 6 8 10 > 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 261]][a, z]
Out[8]=	3 5 3 -2 a 3 a a z 3 5 2 z 3 3 3 ---- + ---- - -- - - - 2 a z + 6 a z - 3 a z - ---- + 2 a z + 7 a z - z z z a a 5 5 3 z 5 3 5 5 5 7 3 7 > 3 a z - -- + 3 a z + 4 a z - a z + a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 261]][a, z]
Out[9]=	3 5 2 4 6 2 a 3 a a 2 z 3 5 2 -3 a - 3 a - a + --- + ---- + -- + --- - 3 a z - 11 a z - 6 a z + z - z z z a 2 3 3 z 2 2 4 2 6 2 8 2 z 7 z 3 > -- + 8 a z + 14 a z + 6 a z - 2 a z + -- - ---- - 4 a z + 2 3 a a a 4 3 3 5 3 7 3 4 5 z 2 4 4 4 > 22 a z + 13 a z - 5 a z - 5 z + ---- - 20 a z - 25 a z - 2 a 5 5 6 4 8 4 z 13 z 5 3 5 5 5 7 5 > 12 a z + 3 a z - -- + ----- + 8 a z - 27 a z - 13 a z + 8 a z + 3 a a 6 7 6 4 z 2 6 4 6 6 6 8 6 8 z 7 > 13 z - ---- + 25 a z + 22 a z + 13 a z - a z - ---- + 2 a z + 2 a a 3 7 5 7 7 7 8 2 8 4 8 6 8 > 24 a z + 11 a z - 3 a z - 9 z - 10 a z - 6 a z - 5 a z - 9 3 9 5 9 2 10 4 10 > 6 a z - 11 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 2 1 5 2 7 5 10 11 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 7 11 10 9 11 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 6 t + 8 q t + 3 q t + 6 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a261

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