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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-17/2 - 4q-15/2 + 7q-13/2 - 11q-11/2 + 15q-9/2 - 17q-7/2 + 16q-5/2 - 15q-3/2 + 10q-1/2 - 7q1/2 + 4q3/2 - q5/2

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

az - 3az3 - 4az5 - az7 - a3z-1 - 5a3z + 4a3z3 + 11a3z5 + 6a3z7 + a3z9 + a5z-1 + 2a5z - 3a5z3 - 4a5z5 - a5z7

Kauffman Polynomial:

- a-1z3 + 3a-1z5 - a-1z7 + 2z2 - 13z4 + 15z6 - 4z8 + az + 9az3 - 27az5 + 24az7 - 6az9 + 6a2z2 - 27a2z4 + 19a2z6 + 3a2z8 - 3a2z10 - a3z-1 + 4a3z + 17a3z3 - 54a3z5 + 48a3z7 - 13a3z9 + a4 + 2a4z2 - 21a4z4 + 21a4z6 - a4z8 - 3a4z10 - a5z-1 + a5z + 7a5z3 - 13a5z5 + 15a5z7 - 7a5z9 - 3a6z2 + a6z4 + 10a6z6 - 8a6z8 - 2a7z + 3a7z3 + 7a7z5 - 8a7z7 - a8z2 + 7a8z4 - 7a8z6 + 3a9z3 - 4a9z5 - 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                     3   j = 2                   4 1   j = 0                 6 3     j = -2               9 4       j = -4             8 7         j = -6           9 8           j = -8         6 8             j = -10       5 9               j = -12     3 7                 j = -14   1 4                   j = -16   3                     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, 250]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 250]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[14, 6, 15, 5], > X[22, 18, 9, 17], X[4, 19, 5, 20], X[6, 22, 7, 21], X[16, 7, 17, 8], > X[8, 9, 1, 10], X[18, 14, 19, 13], X[20, 15, 21, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -6, 4, -7, 8, -9}, > {9, -1, 2, -3, 10, -4, 11, -8, 5, -10, 6, -11, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 4 7 11 15 17 16 15 10 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 > 7 Sqrt[q] + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 -20 3 -14 2 3 6 3 -2 2 4 6 -2 - q + --- - q + --- - q + --- - --- + -- + -- + q + q - 2 q + q 22 18 12 10 8 4 q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 250]][a, z]
Out[8]=	3 5 a a 3 5 3 3 3 5 3 5 -(--) + -- + a z - 5 a z + 2 a z - 3 a z + 4 a z - 3 a z - 4 a z + z z 3 5 5 5 7 3 7 5 7 3 9 > 11 a z - 4 a z - a z + 6 a z - a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 250]][a, z]
Out[9]=	3 5 4 a a 3 5 7 2 2 2 4 2 a - -- - -- + a z + 4 a z + a z - 2 a z + 2 z + 6 a z + 2 a z - z z 3 6 2 8 2 z 3 3 3 5 3 7 3 9 3 > 3 a z - a z - -- + 9 a z + 17 a z + 7 a z + 3 a z + 3 a z - a 5 4 2 4 4 4 6 4 8 4 10 4 3 z 5 > 13 z - 27 a z - 21 a z + a z + 7 a z - a z + ---- - 27 a z - a 3 5 5 5 7 5 9 5 6 2 6 4 6 > 54 a z - 13 a z + 7 a z - 4 a z + 15 z + 19 a z + 21 a z + 7 6 6 8 6 z 7 3 7 5 7 7 7 8 > 10 a z - 7 a z - -- + 24 a z + 48 a z + 15 a z - 8 a z - 4 z + a 2 8 4 8 6 8 9 3 9 5 9 2 10 > 3 a z - a z - 8 a z - 6 a z - 13 a z - 7 a z - 3 a z - 4 10 > 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 9 1 3 1 4 3 7 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 9 6 8 9 8 8 4 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 6 t + --- + 3 t + 4 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 + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a250

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