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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-17/2 - 3q-15/2 + 5q-13/2 - 8q-11/2 + 10q-9/2 - 12q-7/2 + 11q-5/2 - 10q-3/2 + 7q-1/2 - 5q1/2 + 3q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-24 + q-22 - q-20 + 2q-18 + q-16 + q-14 + 3q-12 - q-10 + 4q-8 - q-6 + q-4 - 1 + q2 - q4 + q6

HOMFLY-PT Polynomial:

- 3az - 7az3 - 5az5 - az7 - a3z-1 + 3a3z + 16a3z3 + 17a3z5 + 7a3z7 + a3z9 + a5z-1 - 2a5z - 7a5z3 - 5a5z5 - a5z7

Kauffman Polynomial:

- 3a-1z3 + 4a-1z5 - a-1z7 + 4z2 - 14z4 + 13z6 - 3z8 - 4az + 14az3 - 24az5 + 18az7 - 4az9 + 6a2z2 - 16a2z4 + 8a2z6 + 4a2z8 - 2a2z10 - a3z-1 - 4a3z + 30a3z3 - 51a3z5 + 35a3z7 - 8a3z9 + a4 + 3a4z2 - 10a4z4 + 5a4z6 + 3a4z8 - 2a4z10 - a5z-1 - a5z + 11a5z3 - 17a5z5 + 12a5z7 - 4a5z9 - 3a6z4 + 6a6z6 - 4a6z8 - 2a7z + 2a7z3 + 3a7z5 - 4a7z7 + 4a8z4 - 4a8z6 - a9z + 4a9z3 - 3a9z5 + a10z2 - a10z4

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 6                       1 j = 4                     2   j = 2                   3 1   j = 0                 4 2     j = -2               6 3       j = -4             6 5         j = -6           6 5           j = -8         4 6             j = -10       4 6               j = -12     2 5                 j = -14   1 3                   j = -16   2                     j = -18 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, 253]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 253]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[14, 5, 15, 6], > X[18, 10, 19, 9], X[22, 18, 9, 17], X[8, 21, 1, 22], X[20, 15, 21, 16], > X[16, 8, 17, 7], X[4, 13, 5, 14], X[6, 20, 7, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -10, 4, -11, 9, -7}, > {5, -1, 2, -3, 10, -4, 8, -9, 6, -5, 11, -8, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 5 8 10 12 11 10 7 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 5/2 > 5 Sqrt[q] + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -20 2 -16 -14 3 -10 4 -6 -4 -1 - q + q - q + --- + q + q + --- - q + -- - q + q + 18 12 8 q q q 2 4 6 > q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 253]][a, z]
Out[8]=	3 5 a a 3 5 3 3 3 5 3 5 -(--) + -- - 3 a z + 3 a z - 2 a z - 7 a z + 16 a z - 7 a z - 5 a z + z z 3 5 5 5 7 3 7 5 7 3 9 > 17 a z - 5 a z - a z + 7 a z - a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 253]][a, z]
Out[9]=	3 5 4 a a 3 5 7 9 2 2 2 a - -- - -- - 4 a z - 4 a z - a z - 2 a z - a z + 4 z + 6 a z + z z 3 4 2 10 2 3 z 3 3 3 5 3 7 3 > 3 a z + a z - ---- + 14 a z + 30 a z + 11 a z + 2 a z + a 5 9 3 4 2 4 4 4 6 4 8 4 10 4 4 z > 4 a z - 14 z - 16 a z - 10 a z - 3 a z + 4 a z - a z + ---- - a 5 3 5 5 5 7 5 9 5 6 2 6 > 24 a z - 51 a z - 17 a z + 3 a z - 3 a z + 13 z + 8 a z + 7 4 6 6 6 8 6 z 7 3 7 5 7 > 5 a z + 6 a z - 4 a z - -- + 18 a z + 35 a z + 12 a z - a 7 7 8 2 8 4 8 6 8 9 3 9 5 9 > 4 a z - 3 z + 4 a z + 3 a z - 4 a z - 4 a z - 8 a z - 4 a z - 2 10 4 10 > 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 6 1 2 1 3 2 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 6 4 6 6 5 6 3 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 4 t + --- + 2 t + 3 q t + 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q 2 3 4 3 6 4 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a253

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