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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 3q-17/2 - 8q-15/2 + 12q-13/2 - 17q-11/2 + 19q-9/2 - 19q-7/2 + 17q-5/2 - 12q-3/2 + 7q-1/2 - 4q1/2 + q3/2

A2 (sl(3)) Invariant:

q-30 + q-28 - q-26 + 3q-24 + 2q-22 + 5q-18 - 2q-16 + q-14 - q-12 - 3q-10 + 3q-8 - 4q-6 + 3q-4 + q-2 - 1 + 2q2 - q4

HOMFLY-PT Polynomial:

2az3 + az5 - 5a3z - 7a3z3 - 4a3z5 - a3z7 + 5a5z + 8a5z3 + 3a5z5 - a7z-1 - 5a7z - 3a7z3 + a9z-1 + a9z

Kauffman Polynomial:

2z4 - z6 - 7az3 + 11az5 - 4az7 + 4a2z2 - 13a2z4 + 16a2z6 - 6a2z8 + 4a3z - 18a3z3 + 17a3z5 + a3z7 - 4a3z9 + 16a4z2 - 41a4z4 + 40a4z6 - 13a4z8 - a4z10 + a5z - 10a5z3 + 8a5z5 + 8a5z7 - 8a5z9 + 9a6z2 - 27a6z4 + 31a6z6 - 13a6z8 - a6z10 - a7z-1 + 5a7z - 12a7z3 + 14a7z5 - 3a7z7 - 4a7z9 + a8 - 4a8z2 + 3a8z4 + 5a8z6 - 6a8z8 - a9z-1 + 7a9z - 11a9z3 + 11a9z5 - 6a9z7 - a10z2 + 4a10z4 - 3a10z6 - a11z + 2a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                       1 j = 2                     3   j = 0                   4 1   j = -2                 8 3     j = -4               10 5       j = -6             9 7         j = -8           10 10           j = -10         7 9             j = -12       5 10               j = -14     3 7                 j = -16     5                   j = -18 1 3                     j = -20 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, 307]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 307]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[4, 9, 5, 10], > X[16, 5, 17, 6], X[22, 14, 9, 13], X[14, 18, 15, 17], X[20, 7, 21, 8], > X[18, 22, 19, 21], X[6, 15, 7, 16], X[8, 19, 1, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -4, 5, -10, 8, -11}, > {4, -1, 2, -3, 6, -7, 10, -5, 7, -9, 11, -8, 9, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 8 12 17 19 19 17 12 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 7 3/2 > ------- - 4 Sqrt[q] + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 -26 3 2 5 2 -14 -12 3 3 4 -1 + q + q - q + --- + --- + --- - --- + q - q - --- + -- - -- + 24 22 18 16 10 8 6 q q q q q q q 3 -2 2 4 > -- + q + 2 q - q 4 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 307]][a, z]
Out[8]=	7 9 a a 3 5 7 9 3 3 3 5 3 -(--) + -- - 5 a z + 5 a z - 5 a z + a z + 2 a z - 7 a z + 8 a z - z z 7 3 5 3 5 5 5 3 7 > 3 a z + a z - 4 a z + 3 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 307]][a, z]
Out[9]=	7 9 8 a a 3 5 7 9 11 2 2 4 2 a - -- - -- + 4 a z + a z + 5 a z + 7 a z - a z + 4 a z + 16 a z + z z 6 2 8 2 10 2 3 3 3 5 3 7 3 > 9 a z - 4 a z - a z - 7 a z - 18 a z - 10 a z - 12 a z - 9 3 11 3 4 2 4 4 4 6 4 8 4 > 11 a z + 2 a z + 2 z - 13 a z - 41 a z - 27 a z + 3 a z + 10 4 5 3 5 5 5 7 5 9 5 11 5 > 4 a z + 11 a z + 17 a z + 8 a z + 14 a z + 11 a z - a z - 6 2 6 4 6 6 6 8 6 10 6 7 3 7 > z + 16 a z + 40 a z + 31 a z + 5 a z - 3 a z - 4 a z + a z + 5 7 7 7 9 7 2 8 4 8 6 8 8 8 > 8 a z - 3 a z - 6 a z - 6 a z - 13 a z - 13 a z - 6 a z - 3 9 5 9 7 9 4 10 6 10 > 4 a z - 8 a z - 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 8 1 1 3 5 3 7 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 8 18 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 10 7 9 10 10 9 7 10 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 4 t + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q t q t q t q t q t q t q t q t 3 t 2 2 2 4 3 > --- + t + 3 q t + q t 2 q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a307

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