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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 4q-11/2 - 10q-9/2 + 16q-7/2 - 22q-5/2 + 26q-3/2 - 26q-1/2 + 22q1/2 - 17q3/2 + 10q5/2 - 5q7/2 + q9/2

A2 (sl(3)) Invariant:

q-20 - q-18 + 5q-14 - 3q-12 + 3q-10 + q-8 - 5q-6 + 3q-4 - 5q-2 + 4 - q4 + 6q6 - 2q8 + 2q10 + 2q12 - q14

HOMFLY-PT Polynomial:

- a-3z-1 + a-3z3 + a-1z-1 - 3a-1z3 - 2a-1z5 + az-1 + 3az + 5az3 + 3az5 + az7 - 2a3z-1 - 4a3z - 4a3z3 - 2a3z5 + a5z-1 + a5z + a5z3

Kauffman Polynomial:

a-4z4 - a-4z6 + a-3z-1 - a-3z - 5a-3z3 + 10a-3z5 - 5a-3z7 - a-2 + 2a-2z2 - 11a-2z4 + 19a-2z6 - 9a-2z8 + a-1z-1 + a-1z - 18a-1z3 + 22a-1z5 + 2a-1z7 - 7a-1z9 - 2 + 10z2 - 35z4 + 52z6 - 20z8 - 2z10 - az-1 + 12az - 36az3 + 32az5 + 9az7 - 14az9 - 3a2 + 12a2z2 - 36a2z4 + 51a2z6 - 22a2z8 - 2a2z10 - 2a3z-1 + 15a3z - 35a3z3 + 35a3z5 - 7a3z7 - 7a3z9 - a4 + 3a4z2 - 9a4z4 + 15a4z6 - 11a4z8 - a5z-1 + 5a5z - 11a5z3 + 14a5z5 - 9a5z7 - a6z2 + 4a6z4 - 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                     4   j = 6                   6 1   j = 4                 11 4     j = 2               11 6       j = 0             15 11         j = -2           13 13           j = -4         9 13             j = -6       7 13               j = -8     3 9                 j = -10   1 7                   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, 46]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 46]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[20, 17, 21, 18], X[18, 10, 19, 9], > X[8, 20, 9, 19], 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]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 8, -6}, {9, -1, 2, -5, 4, -8, 7, -9, 10, -7, 11, -2, 3, -4, > 5, -3, 6, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 4 10 16 22 26 26 -q + ----- - ---- + ---- - ---- + ---- - ------- + 22 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 > 17 q + 10 q - 5 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 5 3 3 -8 5 3 5 4 6 8 4 + q - q + --- - --- + --- + q - -- + -- - -- - q + 6 q - 2 q + 14 12 10 6 4 2 q q q q q q 10 12 14 > 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 46]][a, z]
Out[8]=	3 5 3 3 1 1 a 2 a a 3 5 z 3 z 3 -(----) + --- + - - ---- + -- + 3 a z - 4 a z + a z + -- - ---- + 5 a z - 3 a z z z z 3 a a z a 5 3 3 5 3 2 z 5 3 5 7 > 4 a z + a z - ---- + 3 a z - 2 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 46]][a, z]
Out[9]=	3 5 -2 2 4 1 1 a 2 a a z z 3 -2 - a - 3 a - a + ---- + --- - - - ---- - -- - -- + - + 12 a z + 15 a z + 3 a z z z z 3 a a z a 2 3 3 5 2 2 z 2 2 4 2 6 2 5 z 18 z > 5 a z + 10 z + ---- + 12 a z + 3 a z - a z - ---- - ----- - 2 3 a a a 4 4 3 3 3 5 3 7 3 4 z 11 z 2 4 > 36 a z - 35 a z - 11 a z + a z - 35 z + -- - ----- - 36 a z - 4 2 a a 5 5 4 4 6 4 10 z 22 z 5 3 5 5 5 7 5 > 9 a z + 4 a z + ----- + ----- + 32 a z + 35 a z + 14 a z - a z + 3 a a 6 6 7 7 6 z 19 z 2 6 4 6 6 6 5 z 2 z 7 > 52 z - -- + ----- + 51 a z + 15 a z - 4 a z - ---- + ---- + 9 a z - 4 2 3 a a a a 8 9 3 7 5 7 8 9 z 2 8 4 8 7 z 9 > 7 a z - 9 a z - 20 z - ---- - 22 a z - 11 a z - ---- - 14 a z - 2 a a 3 9 10 2 10 > 7 a z - 2 z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	13 1 3 1 7 3 9 7 13 15 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 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 9 13 13 2 2 2 4 2 4 3 > ----- + ---- + ---- + 11 t + 11 q t + 6 q t + 11 q t + 4 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 + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a46

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