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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-11 + 3q-10 - 8q-9 + 13q-8 - 16q-7 + 19q-6 - 17q-5 + 16q-4 - 10q-3 + 6q-2 - 2q-1 + 1

A2 (sl(3)) Invariant:

- q-34 - q-32 - 5q-28 + 8q-20 + 3q-18 + 9q-16 + 4q-14 + 2q-12 + 5q-10 - 2q-8 + 3q-6 + q-4 + 1

HOMFLY-PT Polynomial:

2a2 + 3a2z2 + a2z4 + 2a4z-2 + 2a4 - 2a4z4 - a4z6 - 5a6z-2 - 10a6 - 10a6z2 - 7a6z4 - 2a6z6 + 4a8z-2 + 8a8 + 8a8z2 + 3a8z4 - a10z-2 - 2a10 - a10z2

Kauffman Polynomial:

- 2a2 + 5a2z2 - 4a2z4 + a2z6 + a3z + 2a3z3 - 5a3z5 + 2a3z7 - 2a4z-2 + 6a4 - 6a4z2 + 4a4z4 - 6a4z6 + 3a4z8 + 5a5z-1 - 15a5z + 14a5z3 - 5a5z5 - 3a5z7 + 3a5z9 - 5a6z-2 + 22a6 - 44a6z2 + 49a6z4 - 32a6z6 + 9a6z8 + a6z10 + 9a7z-1 - 35a7z + 43a7z3 - 18a7z5 - 6a7z7 + 7a7z9 - 4a8z-2 + 21a8 - 45a8z2 + 55a8z4 - 39a8z6 + 13a8z8 + a8z10 + 5a9z-1 - 25a9z + 41a9z3 - 29a9z5 + 5a9z7 + 4a9z9 - a10z-2 + 6a10 - 11a10z2 + 10a10z4 - 11a10z6 + 7a10z8 + a11z-1 - 5a11z + 8a11z3 - 10a11z5 + 6a11z7 + a12z2 - 4a12z4 + 3a12z6 + a13z - 2a13z3 + a13z5

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                       1 j = -1                     1   j = -3                   5 1   j = -5                 7 3     j = -7               9 3       j = -9             8 7         j = -11           11 9           j = -13         7 10             j = -15       6 9               j = -17     2 7                 j = -19   1 6                   j = -21   2                     j = -23 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, 491]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 491]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 11, 17, 12], X[20, 8, 21, 7], > X[8, 20, 9, 19], X[14, 21, 15, 22], X[22, 15, 19, 16], X[18, 13, 5, 14], > X[12, 17, 13, 18], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {5, -4, 6, -7}, > {10, -1, 4, -5, 11, -2, 3, -9, 8, -6, 7, -3, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-11 3 8 13 16 19 17 16 10 6 2 1 - q + --- - -- + -- - -- + -- - -- + -- - -- + -- - - 10 9 8 7 6 5 4 3 2 q q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -32 5 8 3 9 4 2 5 2 3 -4 1 - q - q - --- + --- + --- + --- + --- + --- + --- - -- + -- + q 28 20 18 16 14 12 10 8 6 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 491]][a, z]
Out[8]=	4 6 8 10 2 4 6 8 10 2 a 5 a 4 a a 2 2 2 a + 2 a - 10 a + 8 a - 2 a + ---- - ---- + ---- - --- + 3 a z - 2 2 2 2 z z z z 6 2 8 2 10 2 2 4 4 4 6 4 8 4 4 6 > 10 a z + 8 a z - a z + a z - 2 a z - 7 a z + 3 a z - a z - 6 6 > 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 491]][a, z]
Out[9]=	4 6 8 10 5 7 2 4 6 8 10 2 a 5 a 4 a a 5 a 9 a -2 a + 6 a + 22 a + 21 a + 6 a - ---- - ---- - ---- - --- + ---- + ---- + 2 2 2 2 z z z z z z 9 11 5 a a 3 5 7 9 11 13 > ---- + --- + a z - 15 a z - 35 a z - 25 a z - 5 a z + a z + z z 2 2 4 2 6 2 8 2 10 2 12 2 3 3 > 5 a z - 6 a z - 44 a z - 45 a z - 11 a z + a z + 2 a z + 5 3 7 3 9 3 11 3 13 3 2 4 4 4 > 14 a z + 43 a z + 41 a z + 8 a z - 2 a z - 4 a z + 4 a z + 6 4 8 4 10 4 12 4 3 5 5 5 7 5 > 49 a z + 55 a z + 10 a z - 4 a z - 5 a z - 5 a z - 18 a z - 9 5 11 5 13 5 2 6 4 6 6 6 8 6 > 29 a z - 10 a z + a z + a z - 6 a z - 32 a z - 39 a z - 10 6 12 6 3 7 5 7 7 7 9 7 11 7 > 11 a z + 3 a z + 2 a z - 3 a z - 6 a z + 5 a z + 6 a z + 4 8 6 8 8 8 10 8 5 9 7 9 9 9 > 3 a z + 9 a z + 13 a z + 7 a z + 3 a z + 7 a z + 4 a z + 6 10 8 10 > a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 2 1 6 2 7 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 23 9 21 8 19 8 19 7 17 7 17 6 15 6 q q q t q t q t q t q t q t q t 9 7 10 11 9 8 7 9 3 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 15 5 13 5 13 4 11 4 11 3 9 3 9 2 7 2 7 q t q t q t q t q t q t q t q t q t 7 t t 2 > ---- + -- + - + q t 5 3 q q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a491

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