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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X18,13,5,14 X14,7,15,8 X10,15,11,16 X22,18,19,17 X20,10,21,9 X8,20,9,19 X16,22,17,21 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-7 + 3q-6 - 7q-5 + 12q-4 - 16q-3 + 21q-2 - 20q-1 + 19 - 14q + 10q2 - 4q3 + q4

A2 (sl(3)) Invariant:

- q-22 - q-20 - 4q-16 + q-14 + q-12 - q-10 + 8q-8 + 3q-6 + 7q-4 + 5q-2 + 1 + 6q2 - 2q4 + 3q6 + 2q8 - 2q10 + q12

HOMFLY-PT Polynomial:

a-2 + a-2z2 + a-2z4 + 2z-2 + 4 + 2z2 - z4 - z6 - 5a2z-2 - 13a2 - 15a2z2 - 8a2z4 - 2a2z6 + 4a4z-2 + 10a4 + 9a4z2 + 3a4z4 - a6z-2 - 2a6 - a6z2

Kauffman Polynomial:

a-4z4 + 4a-3z5 - 2a-2 + 6a-2z2 - 10a-2z4 + 10a-2z6 + 6a-1z3 - 18a-1z5 + 14a-1z7 - 2z-2 + 7 - 10z2 + 9z4 - 18z6 + 13z8 + 5az-1 - 22az + 44az3 - 48az5 + 10az7 + 6az9 - 5a2z-2 + 21a2 - 43a2z2 + 66a2z4 - 66a2z6 + 22a2z8 + a2z10 + 9a3z-1 - 39a3z + 66a3z3 - 39a3z5 - 8a3z7 + 9a3z9 - 4a4z-2 + 16a4 - 35a4z2 + 60a4z4 - 49a4z6 + 12a4z8 + a4z10 + 5a5z-1 - 21a5z + 34a5z3 - 17a5z5 - 3a5z7 + 3a5z9 - a6z-2 + 3a6 - 8a6z2 + 14a6z4 - 11a6z6 + 3a6z8 + a7z-1 - 4a7z + 6a7z3 - 4a7z5 + a7z7

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 = 9                       1 j = 7                     4 1 j = 5                   6     j = 3                 8 4     j = 1               11 6       j = -1             11 10         j = -3           10 9           j = -5         6 11             j = -7       6 10               j = -9     2 7                 j = -11   1 5                   j = -13   2                     j = -15 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 498]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 498]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[18, 13, 5, 14], X[14, 7, 15, 8], > X[10, 15, 11, 16], X[22, 18, 19, 17], X[20, 10, 21, 9], X[8, 20, 9, 19], > X[16, 22, 17, 21], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {8, -7, 9, -6}, > {10, -1, 4, -8, 7, -5, 11, -2, 3, -4, 5, -9, 6, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-7 3 7 12 16 21 20 2 3 4 19 - q + -- - -- + -- - -- + -- - -- - 14 q + 10 q - 4 q + q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 4 -14 -12 -10 8 3 7 5 2 4 1 - q - q - --- + q + q - q + -- + -- + -- + -- + 6 q - 2 q + 16 8 6 4 2 q q q q q 6 8 10 12 > 3 q + 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 498]][a, z]
Out[8]=	2 4 6 2 -2 2 4 6 2 5 a 4 a a 2 z 2 2 4 + a - 13 a + 10 a - 2 a + -- - ---- + ---- - -- + 2 z + -- - 15 a z + 2 2 2 2 2 z z z z a 4 4 2 6 2 4 z 2 4 4 4 6 2 6 > 9 a z - a z - z + -- - 8 a z + 3 a z - z - 2 a z 2 a
In[9]:=	Kauffman[Link[11, Alternating, 498]][a, z]
Out[9]=	2 4 6 3 5 2 2 4 6 2 5 a 4 a a 5 a 9 a 5 a 7 - -- + 21 a + 16 a + 3 a - -- - ---- - ---- - -- + --- + ---- + ---- + 2 2 2 2 2 z z z a z z z z 7 2 a 3 5 7 2 6 z 2 2 > -- - 22 a z - 39 a z - 21 a z - 4 a z - 10 z + ---- - 43 a z - z 2 a 3 4 2 6 2 6 z 3 3 3 5 3 7 3 > 35 a z - 8 a z + ---- + 44 a z + 66 a z + 34 a z + 6 a z + a 4 4 5 5 4 z 10 z 2 4 4 4 6 4 4 z 18 z > 9 z + -- - ----- + 66 a z + 60 a z + 14 a z + ---- - ----- - 4 2 3 a a a a 6 5 3 5 5 5 7 5 6 10 z 2 6 > 48 a z - 39 a z - 17 a z - 4 a z - 18 z + ----- - 66 a z - 2 a 7 4 6 6 6 14 z 7 3 7 5 7 7 7 8 > 49 a z - 11 a z + ----- + 10 a z - 8 a z - 3 a z + a z + 13 z + a 2 8 4 8 6 8 9 3 9 5 9 2 10 4 10 > 22 a z + 12 a z + 3 a z + 6 a z + 9 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 2 1 5 2 7 6 10 -- + 11 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 7 3 q t q t q t q t q t q t q t q t 6 11 10 9 11 3 3 2 5 2 > ----- + ----- + ----- + ---- + --- + 6 q t + 8 q t + 4 q t + 6 q t + 5 3 5 2 3 2 3 q t q t q t q t q t 7 3 7 4 9 4 > 4 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a498

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