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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-21/2 + 2q-19/2 - 3q-17/2 + 4q-15/2 - 5q-13/2 + 5q-11/2 - 5q-9/2 + 4q-7/2 - 4q-5/2 + 2q-3/2 - 2q-1/2 + q1/2

A2 (sl(3)) Invariant:

q-30 + q-26 + 2q-16 + 2q-12 + q-10 + 2q-8 + q-6 - 1

HOMFLY-PT Polynomial:

4a3z + 10a3z3 + 6a3z5 + a3z7 - a5z-1 - 14a5z - 29a5z3 - 23a5z5 - 8a5z7 - a5z9 + a7z-1 + 7a7z + 11a7z3 + 6a7z5 + a7z7

Kauffman Polynomial:

4a2z2 - 10a2z4 + 6a2z6 - a2z8 - 5a3z + 21a3z3 - 27a3z5 + 13a3z7 - 2a3z9 + 6a4z2 - 10a4z4 + 4a4z8 - a4z10 + a5z-1 - 15a5z + 42a5z3 - 48a5z5 + 24a5z7 - 4a5z9 - a6 + 7a6z2 - 9a6z4 + 2a6z6 + 3a6z8 - a6z10 + a7z-1 - 8a7z + 16a7z3 - 15a7z5 + 9a7z7 - 2a7z9 + 2a8z2 - 5a8z4 + 6a8z6 - 2a8z8 + a9z - 3a9z3 + 4a9z5 - 2a9z7 - a10z2 + 2a10z4 - 2a10z6 + a11z3 - 2a11z5 + 2a12z2 - 2a12z4 + a13z - a13z3

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 = 2                       1 j = 0                     1   j = -2                   1 1   j = -4                 3 1     j = -6               2 2       j = -8             3 2         j = -10           2 2           j = -12         3 3             j = -14       1 2               j = -16     2 3                 j = -18   1 2                   j = -20   1                     j = -22 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, 367]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 367]]
Out[4]=	PD[X[12, 1, 13, 2], X[2, 13, 3, 14], X[14, 3, 15, 4], X[16, 5, 17, 6], > X[18, 7, 19, 8], X[8, 11, 9, 12], X[20, 10, 21, 9], X[22, 20, 11, 19], > X[10, 22, 1, 21], X[4, 15, 5, 16], X[6, 17, 7, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -10, 4, -11, 5, -6, 7, -9}, > {6, -1, 2, -3, 10, -4, 11, -5, 8, -7, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 2 3 4 5 5 5 4 4 2 -q + ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- + ---- - 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q 2 > ------- + Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -26 2 2 -10 2 -6 -1 + q + q + --- + --- + q + -- + q 16 12 8 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 367]][a, z]
Out[8]=	5 7 a a 3 5 7 3 3 5 3 7 3 -(--) + -- + 4 a z - 14 a z + 7 a z + 10 a z - 29 a z + 11 a z + z z 3 5 5 5 7 5 3 7 5 7 7 7 5 9 > 6 a z - 23 a z + 6 a z + a z - 8 a z + a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 367]][a, z]
Out[9]=	5 7 6 a a 3 5 7 9 13 2 2 4 2 -a + -- + -- - 5 a z - 15 a z - 8 a z + a z + a z + 4 a z + 6 a z + z z 6 2 8 2 10 2 12 2 3 3 5 3 7 3 > 7 a z + 2 a z - a z + 2 a z + 21 a z + 42 a z + 16 a z - 9 3 11 3 13 3 2 4 4 4 6 4 8 4 > 3 a z + a z - a z - 10 a z - 10 a z - 9 a z - 5 a z + 10 4 12 4 3 5 5 5 7 5 9 5 11 5 > 2 a z - 2 a z - 27 a z - 48 a z - 15 a z + 4 a z - 2 a z + 2 6 6 6 8 6 10 6 3 7 5 7 7 7 > 6 a z + 2 a z + 6 a z - 2 a z + 13 a z + 24 a z + 9 a z - 9 7 2 8 4 8 6 8 8 8 3 9 5 9 > 2 a z - a z + 4 a z + 3 a z - 2 a z - 2 a z - 4 a z - 7 9 4 10 6 10 > 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 1 1 2 2 3 1 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 22 8 20 7 18 7 18 6 16 6 16 5 14 5 q q q t q t q t q t q t q t q t 2 3 3 2 2 3 2 2 t > ------ + ------ + ------ + ------ + ------ + ----- + ---- + ---- + -- + 14 4 12 4 12 3 10 3 10 2 8 2 8 6 4 q t q t q t q t q t q t q t q t q 2 t 2 t 2 3 > -- + t + -- + q t 2 2 q q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a367

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