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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-21/2 - 2q-19/2 + 4q-17/2 - 8q-15/2 + 10q-13/2 - 12q-11/2 + 12q-9/2 - 11q-7/2 + 8q-5/2 - 5q-3/2 + 2q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-32 - q-26 + 3q-24 + 2q-18 - q-16 + 3q-14 + q-10 + 2q-8 - 2q-6 + 2q-4 + q2

HOMFLY-PT Polynomial:

- 2az - az3 - a3z-1 - 2a3z + a3z3 + a3z5 + a5z-1 + 3a5z + 5a5z3 + 2a5z5 + a7z + 2a7z3 + a7z5 - 2a9z - a9z3

Kauffman Polynomial:

- 2az + 3az3 - az5 - a2z2 + 4a2z4 - 2a2z6 - a3z-1 + 4a3z - 5a3z3 + 6a3z5 - 3a3z7 + a4 - 4a4z4 + 5a4z6 - 3a4z8 - a5z-1 + 5a5z - 9a5z3 + 3a5z5 + a5z7 - 2a5z9 - 3a6z2 + 3a6z4 - 2a6z6 - a6z10 - 4a7z + 15a7z3 - 19a7z5 + 11a7z7 - 4a7z9 - a8z2 + 7a8z4 - 4a8z6 + a8z8 - a8z10 - a9z + 9a9z3 - 8a9z5 + 5a9z7 - 2a9z9 - a10z2 + 4a10z6 - 2a10z8 + 2a11z - 7a11z3 + 7a11z5 - 2a11z7 - 4a12z2 + 4a12z4 - 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                     1   j = -2                   4 1   j = -4                 5 2     j = -6               6 3       j = -8             6 5         j = -10           6 6           j = -12         5 7             j = -14       3 5               j = -16     1 5                 j = -18   1 3                   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, 260]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 260]]
Out[4]=	PD[X[10, 1, 11, 2], X[14, 5, 15, 6], X[12, 3, 13, 4], X[18, 8, 19, 7], > X[22, 15, 9, 16], X[20, 17, 21, 18], X[16, 21, 17, 22], X[4, 13, 5, 14], > X[6, 20, 7, 19], X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 3, -8, 2, -9, 4, -11}, > {10, -1, 11, -3, 8, -2, 5, -7, 6, -4, 9, -6, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 2 4 8 10 12 12 11 8 5 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]=	-32 -26 3 2 -16 3 -10 2 2 2 2 -q - q + --- + --- - q + --- + q + -- - -- + -- + q 24 18 14 8 6 4 q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 260]][a, z]
Out[8]=	3 5 a a 3 5 7 9 3 3 3 5 3 -(--) + -- - 2 a z - 2 a z + 3 a z + a z - 2 a z - a z + a z + 5 a z + z z 7 3 9 3 3 5 5 5 7 5 > 2 a z - a z + a z + 2 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 260]][a, z]
Out[9]=	3 5 4 a a 3 5 7 9 11 2 2 a - -- - -- - 2 a z + 4 a z + 5 a z - 4 a z - a z + 2 a z - a z - z z 6 2 8 2 10 2 12 2 3 3 3 5 3 > 3 a z - a z - a z - 4 a z + 3 a z - 5 a z - 9 a z + 7 3 9 3 11 3 2 4 4 4 6 4 8 4 > 15 a z + 9 a z - 7 a z + 4 a z - 4 a z + 3 a z + 7 a z + 12 4 5 3 5 5 5 7 5 9 5 11 5 > 4 a z - a z + 6 a z + 3 a z - 19 a z - 8 a z + 7 a z - 2 6 4 6 6 6 8 6 10 6 12 6 3 7 > 2 a z + 5 a z - 2 a z - 4 a z + 4 a z - a z - 3 a z + 5 7 7 7 9 7 11 7 4 8 8 8 10 8 > a z + 11 a z + 5 a z - 2 a z - 3 a z + a z - 2 a z - 5 9 7 9 9 9 6 10 8 10 > 2 a z - 4 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 1 1 1 3 1 5 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 5 5 7 6 6 6 5 6 3 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 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 5 t 2 2 > ---- + t + -- + q t 4 2 q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a260

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