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DrawMorseLink

PD Presentation:

X6172 X14,3,15,4 X16,12,17,11 X12,16,13,15 X22,17,5,18 X18,7,19,8 X8,21,9,22 X20,9,21,10 X10,19,11,20 X2536 X4,13,1,14

Gauss Code:

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

Jones Polynomial:

q-21/2 - 2q-19/2 + 5q-17/2 - 9q-15/2 + 12q-13/2 - 16q-11/2 + 16q-9/2 - 15q-7/2 + 11q-5/2 - 8q-3/2 + 4q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-34 - 2q-32 - q-28 - 2q-26 + 4q-24 + q-22 + q-20 + 5q-18 + 3q-14 - q-12 + 3q-8 - 3q-6 + 3q-4 - 2 + q2

HOMFLY-PT Polynomial:

- az3 - a3z + a3z5 - a5z-1 - 2a5z - a5z3 + a5z5 - 2a7z - 3a7z3 + 2a9z-1 + 3a9z - a11z-1

Kauffman Polynomial:

az3 - az5 + 6a2z4 - 4a2z6 + a3z - 5a3z3 + 13a3z5 - 7a3z7 + 3a4z2 - 9a4z4 + 13a4z6 - 7a4z8 + a5z-1 - 3a5z - 2a5z3 + 4a5z5 + 2a5z7 - 4a5z9 - a6 + 7a6z2 - 23a6z4 + 22a6z6 - 8a6z8 - a6z10 - a7z + 5a7z3 - 12a7z5 + 12a7z7 - 6a7z9 + 3a8 - 4a8z2 - 2a8z4 + 7a8z6 - 3a8z8 - a8z10 - 2a9z-1 + 5a9z - 4a9z3 + 4a9z5 + a9z7 - 2a9z9 + 5a10 - 13a10z2 + 10a10z4 + a10z6 - 2a10z8 - a11z-1 + 2a11z - 5a11z3 + 6a11z5 - 2a11z7 + 2a12 - 5a12z2 + 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                     3   j = -2                   5 1   j = -4                 7 4     j = -6               8 4       j = -8             8 7         j = -10           8 8           j = -12         5 9             j = -14       4 7               j = -16     1 5                 j = -18   1 4                   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, 124]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 124]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[16, 12, 17, 11], X[12, 16, 13, 15], > X[22, 17, 5, 18], X[18, 7, 19, 8], X[8, 21, 9, 22], X[20, 9, 21, 10], > X[10, 19, 11, 20], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 6, -7, 8, -9, 3, -4, 11, -2, 4, -3, 5, -6, > 9, -8, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 2 5 9 12 16 16 15 11 8 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]=	-34 2 -28 2 4 -22 -20 5 3 -12 3 3 -2 - q - --- - q - --- + --- + q + q + --- + --- - q + -- - -- + 32 26 24 18 14 8 6 q q q q q q q 3 2 > -- + q 4 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 124]][a, z]
Out[8]=	5 9 11 a 2 a a 3 5 7 9 3 5 3 7 3 -(--) + ---- - --- - a z - 2 a z - 2 a z + 3 a z - a z - a z - 3 a z + z z z 3 5 5 5 > a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 124]][a, z]
Out[9]=	5 9 11 6 8 10 12 a 2 a a 3 5 7 9 -a + 3 a + 5 a + 2 a + -- - ---- - --- + a z - 3 a z - a z + 5 a z + z z z 11 4 2 6 2 8 2 10 2 12 2 3 > 2 a z + 3 a z + 7 a z - 4 a z - 13 a z - 5 a z + a z - 3 3 5 3 7 3 9 3 11 3 2 4 4 4 > 5 a z - 2 a z + 5 a z - 4 a z - 5 a z + 6 a z - 9 a z - 6 4 8 4 10 4 12 4 5 3 5 5 5 > 23 a z - 2 a z + 10 a z + 4 a z - a z + 13 a z + 4 a z - 7 5 9 5 11 5 2 6 4 6 6 6 8 6 > 12 a z + 4 a z + 6 a z - 4 a z + 13 a z + 22 a z + 7 a z + 10 6 12 6 3 7 5 7 7 7 9 7 11 7 > a z - a z - 7 a z + 2 a z + 12 a z + a z - 2 a z - 4 8 6 8 8 8 10 8 5 9 7 9 9 9 > 7 a z - 8 a z - 3 a z - 2 a z - 4 a z - 6 a z - 2 a z - 6 10 8 10 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 5 1 1 1 4 1 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 7 5 9 8 8 8 7 8 4 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 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 7 t 2 2 > ---- + 3 t + -- + q t 4 2 q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a124

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