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DrawMorseLink

PD Presentation:

X6172 X12,7,13,8 X4,13,1,14 X5,18,6,19 X8493 X20,9,21,10 X10,19,11,20 X17,14,18,15 X15,22,16,17 X21,16,22,5 X2,12,3,11

Gauss Code:

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

Jones Polynomial:

- q-10 + 3q-9 - 4q-8 + 7q-7 - 7q-6 + 8q-5 - 7q-4 + 6q-3 - 3q-2 + 2q-1

A2 (sl(3)) Invariant:

- q-32 - q-30 + 2q-28 + q-26 + 3q-24 + 5q-22 + 2q-20 + 4q-18 + 2q-16 + 2q-14 + 2q-12 + 3q-8 + q-6 + 2q-2

HOMFLY-PT Polynomial:

2a2 + 2a2z2 + a4z-2 + a4 - a4z4 - 2a6z-2 - 6a6 - 5a6z2 - 2a6z4 + a8z-2 + 4a8 + 3a8z2 - a10

Kauffman Polynomial:

- 2a2 + 3a2z2 + a3z + a3z3 + a3z5 - a4z-2 + 3a4 - 3a4z2 + 2a4z6 + 2a5z-1 - 7a5z + 10a5z3 - 10a5z5 + 4a5z7 - 2a6z-2 + 11a6 - 25a6z2 + 28a6z4 - 19a6z6 + 5a6z8 + 2a7z-1 - 9a7z + 17a7z3 - 12a7z5 - 2a7z7 + 2a7z9 - a8z-2 + 7a8 - 27a8z2 + 47a8z4 - 35a8z6 + 8a8z8 - 2a9z + 12a9z3 - 5a9z5 - 5a9z7 + 2a9z9 - 8a10z2 + 19a10z4 - 14a10z6 + 3a10z8 - a11z + 4a11z3 - 4a11z5 + a11z7

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -1                   2 j = -3                 3 2 j = -5               3     j = -7             4 3     j = -9           4 3       j = -11         3 4         j = -13       4 4           j = -15     2 5             j = -17   1 2               j = -19   2                 j = -21 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, NonAlternating, 361]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 361]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 7, 13, 8], X[4, 13, 1, 14], X[5, 18, 6, 19], > X[8, 4, 9, 3], X[20, 9, 21, 10], X[10, 19, 11, 20], X[17, 14, 18, 15], > X[15, 22, 16, 17], X[21, 16, 22, 5], X[2, 12, 3, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-8, 4, 7, -6, -10, 9}, > {-4, -1, 2, -5, 6, -7, 11, -2, 3, 8, -9, 10}]
In[6]:=	Jones[L][q]
Out[6]=	-10 3 4 7 7 8 7 6 3 2 -q + -- - -- + -- - -- + -- - -- + -- - -- + - 9 8 7 6 5 4 3 2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 2 -26 3 5 2 4 2 2 2 3 -q - q + --- + q + --- + --- + --- + --- + --- + --- + --- + -- + 28 24 22 20 18 16 14 12 8 q q q q q q q q q -6 2 > q + -- 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 361]][a, z]
Out[8]=	4 6 8 2 4 6 8 10 a 2 a a 2 2 6 2 8 2 2 a + a - 6 a + 4 a - a + -- - ---- + -- + 2 a z - 5 a z + 3 a z - 2 2 2 z z z 4 4 6 4 > a z - 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 361]][a, z]
Out[9]=	4 6 8 5 7 2 4 6 8 a 2 a a 2 a 2 a 3 5 -2 a + 3 a + 11 a + 7 a - -- - ---- - -- + ---- + ---- + a z - 7 a z - 2 2 2 z z z z z 7 9 11 2 2 4 2 6 2 8 2 > 9 a z - 2 a z - a z + 3 a z - 3 a z - 25 a z - 27 a z - 10 2 3 3 5 3 7 3 9 3 11 3 6 4 > 8 a z + a z + 10 a z + 17 a z + 12 a z + 4 a z + 28 a z + 8 4 10 4 3 5 5 5 7 5 9 5 11 5 > 47 a z + 19 a z + a z - 10 a z - 12 a z - 5 a z - 4 a z + 4 6 6 6 8 6 10 6 5 7 7 7 9 7 > 2 a z - 19 a z - 35 a z - 14 a z + 4 a z - 2 a z - 5 a z + 11 7 6 8 8 8 10 8 7 9 9 9 > a z + 5 a z + 8 a z + 3 a z + 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 2 1 2 2 5 4 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 21 9 19 8 17 8 17 7 15 7 15 6 13 6 q q t q t q t q t q t q t q t 4 3 4 4 3 4 3 3 3 > ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- 13 5 11 5 11 4 9 4 9 3 7 3 7 2 5 2 3 q t q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n361

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