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DrawMorseLink

PD Presentation:

X8192 X16,7,17,8 X2,15,3,16 X18,5,19,6 X12,3,13,4 X22,11,7,12 X4,21,5,22 X14,20,15,19 X20,14,21,13 X6,9,1,10 X10,17,11,18

Gauss Code:

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

Jones Polynomial:

q-21/2 - 5q-19/2 + 11q-17/2 - 18q-15/2 + 24q-13/2 - 28q-11/2 + 27q-9/2 - 24q-7/2 + 17q-5/2 - 10q-3/2 + 4q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-32 + 3q-30 - 5q-26 + 5q-24 - 2q-22 + q-20 + 6q-18 - 2q-16 + 5q-14 - 4q-12 + q-10 + 3q-8 - 5q-6 + 5q-4 - 2 + q2

HOMFLY-PT Polynomial:

- az3 - 3a3z - 2a3z3 + a3z5 - a5z-1 + a5z + 4a5z3 + 3a5z5 + a7z-1 - 2a7z - 2a7z3 + a7z5 + a9z - a9z3

Kauffman Polynomial:

az3 - az5 + 4a2z4 - 4a2z6 + 3a3z - 8a3z3 + 13a3z5 - 9a3z7 + 6a4z2 - 19a4z4 + 23a4z6 - 13a4z8 + a5z-1 - a5z - 5a5z3 - a5z5 + 13a5z7 - 11a5z9 - a6 + 13a6z2 - 44a6z4 + 50a6z6 - 16a6z8 - 4a6z10 + a7z-1 - 3a7z + 8a7z3 - 27a7z5 + 43a7z7 - 21a7z9 + 8a8z2 - 33a8z4 + 45a8z6 - 13a8z8 - 4a8z10 + a9z3 - 3a9z5 + 16a9z7 - 10a9z9 + a10z2 - 11a10z4 + 21a10z6 - 10a10z8 - a11z - 3a11z3 + 9a11z5 - 5a11z7 + a12z4 - a12z6

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 = 2                       1 j = 0                     3   j = -2                   7 1   j = -4                 11 4     j = -6               13 6       j = -8             14 11         j = -10           14 13           j = -12         10 14             j = -14       8 14               j = -16     4 11                 j = -18   1 7                   j = -20   4                     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, 245]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 245]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 7, 17, 8], X[2, 15, 3, 16], X[18, 5, 19, 6], > X[12, 3, 13, 4], X[22, 11, 7, 12], X[4, 21, 5, 22], X[14, 20, 15, 19], > X[20, 14, 21, 13], X[6, 9, 1, 10], X[10, 17, 11, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -3, 5, -7, 4, -10}, > {2, -1, 10, -11, 6, -5, 9, -8, 3, -2, 11, -4, 8, -9, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 5 11 18 24 28 27 24 17 10 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 4 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 3 5 5 2 -20 6 2 5 4 -10 3 -2 - q + --- - --- + --- - --- + q + --- - --- + --- - --- + q + -- - 30 26 24 22 18 16 14 12 8 q q q q q q q q q 5 5 2 > -- + -- + q 6 4 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 245]][a, z]
Out[8]=	5 7 a a 3 5 7 9 3 3 3 5 3 -(--) + -- - 3 a z + a z - 2 a z + a z - a z - 2 a z + 4 a z - z z 7 3 9 3 3 5 5 5 7 5 > 2 a z - a z + a z + 3 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 245]][a, z]
Out[9]=	5 7 6 a a 3 5 7 11 4 2 6 2 8 2 -a + -- + -- + 3 a z - a z - 3 a z - a z + 6 a z + 13 a z + 8 a z + z z 10 2 3 3 3 5 3 7 3 9 3 11 3 2 4 > a z + a z - 8 a z - 5 a z + 8 a z + a z - 3 a z + 4 a z - 4 4 6 4 8 4 10 4 12 4 5 3 5 > 19 a z - 44 a z - 33 a z - 11 a z + a z - a z + 13 a z - 5 5 7 5 9 5 11 5 2 6 4 6 6 6 > a z - 27 a z - 3 a z + 9 a z - 4 a z + 23 a z + 50 a z + 8 6 10 6 12 6 3 7 5 7 7 7 9 7 > 45 a z + 21 a z - a z - 9 a z + 13 a z + 43 a z + 16 a z - 11 7 4 8 6 8 8 8 10 8 5 9 > 5 a z - 13 a z - 16 a z - 13 a z - 10 a z - 11 a z - 7 9 9 9 6 10 8 10 > 21 a z - 10 a z - 4 a z - 4 a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 7 1 4 1 7 4 11 8 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q q t q t q t q t q t q t q t 14 10 14 14 13 14 11 13 6 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 11 t 2 2 > ---- + 3 t + -- + q t 4 2 q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a245

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