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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-17/2 - 3q-15/2 + 7q-13/2 - 12q-11/2 + 15q-9/2 - 18q-7/2 + 17q-5/2 - 15q-3/2 + 11q-1/2 - 7q1/2 + 3q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-26 + q-22 - 2q-20 + 3q-18 - q-16 + 3q-12 - 2q-10 + 5q-8 - q-6 + 2q-4 + 2q-2 - 2 + 2q2 - q4 + q8

HOMFLY-PT Polynomial:

- 2a-1z - a-1z3 - az-1 + az + 5az3 + 2az5 + a3z-1 - 2a3z3 - 3a3z5 - a3z7 + 2a5z + 5a5z3 + 2a5z5 - 2a7z - a7z3

Kauffman Polynomial:

2a-1z - 5a-1z3 + 4a-1z5 - a-1z7 + 4z2 - 12z4 + 11z6 - 3z8 + az-1 - 2az + az3 - 8az5 + 12az7 - 4az9 - a2 + 6a2z2 - 21a2z4 + 21a2z6 - 2a2z8 - 2a2z10 + a3z-1 - 4a3z + 8a3z3 - 19a3z5 + 27a3z7 - 10a3z9 + 7a4z2 - 25a4z4 + 31a4z6 - 8a4z8 - 2a4z10 + 5a5z - 14a5z3 + 11a5z5 + 5a5z7 - 6a5z9 + a6z2 - 9a6z4 + 15a6z6 - 9a6z8 + 5a7z - 14a7z3 + 15a7z5 - 9a7z7 - 3a8z2 + 6a8z4 - 6a8z6 + 2a9z3 - 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                   5 1   j = 0                 6 2     j = -2               9 5       j = -4             9 7         j = -6           9 8           j = -8         7 10             j = -10       5 8               j = -12     2 7                 j = -14   1 5                   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, 380]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 380]]
Out[4]=	PD[X[12, 1, 13, 2], X[2, 13, 3, 14], X[14, 3, 15, 4], X[20, 15, 21, 16], > X[18, 6, 19, 5], X[6, 11, 7, 12], X[22, 7, 11, 8], X[8, 18, 9, 17], > X[16, 10, 17, 9], X[4, 20, 5, 19], X[10, 21, 1, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -10, 5, -6, 7, -8, 9, -11}, > {6, -1, 2, -3, 4, -9, 8, -5, 10, -4, 11, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 7 12 15 18 17 15 11 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] + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -22 2 3 -16 3 2 5 -6 2 2 2 -2 - q + q - --- + --- - q + --- - --- + -- - q + -- + -- + 2 q - 20 18 12 10 8 4 2 q q q q q q q 4 8 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 380]][a, z]
Out[8]=	3 3 a a 2 z 5 7 z 3 3 3 5 3 -(-) + -- - --- + a z + 2 a z - 2 a z - -- + 5 a z - 2 a z + 5 a z - z z a a 7 3 5 3 5 5 5 3 7 > a z + 2 a z - 3 a z + 2 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 380]][a, z]
Out[9]=	3 2 a a 2 z 3 5 7 2 2 2 -a + - + -- + --- - 2 a z - 4 a z + 5 a z + 5 a z + 4 z + 6 a z + z z a 3 4 2 6 2 8 2 10 2 5 z 3 3 3 5 3 > 7 a z + a z - 3 a z + a z - ---- + a z + 8 a z - 14 a z - a 7 3 9 3 4 2 4 4 4 6 4 8 4 > 14 a z + 2 a z - 12 z - 21 a z - 25 a z - 9 a z + 6 a z - 5 10 4 4 z 5 3 5 5 5 7 5 9 5 6 > a z + ---- - 8 a z - 19 a z + 11 a z + 15 a z - 3 a z + 11 z + a 7 2 6 4 6 6 6 8 6 z 7 3 7 > 21 a z + 31 a z + 15 a z - 6 a z - -- + 12 a z + 27 a z + a 5 7 7 7 8 2 8 4 8 6 8 9 > 5 a z - 9 a z - 3 z - 2 a z - 8 a z - 9 a z - 4 a z - 3 9 5 9 2 10 4 10 > 10 a z - 6 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 9 1 2 1 5 2 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 8 7 10 9 8 9 5 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 6 t + --- + 2 t + 5 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 L11a380

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