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L11n373 L11n375

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DrawMorseLink

PD Presentation:

X6172 X2,9,3,10 X12,3,13,4 X10,5,11,6 X16,11,5,12 X4,15,1,16 X13,20,14,21 X7,19,8,18 X17,9,18,8 X19,22,20,17 X21,14,22,15

Gauss Code:

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

Jones Polynomial:

- q-9 + 3q-8 - 5q-7 + 8q-6 - 8q-5 + 8q-4 - 6q-3 + 6q-2 - 2q-1 + 1

A2 (sl(3)) Invariant:

- q-32 + q-26 + 2q-24 - q-22 + 2q-20 + 2q-16 + 4q-14 + 2q-12 + 6q-10 + 3q-8 + 4q-6 + 2q-4 + 1

HOMFLY-PT Polynomial:

a2z-2 + 3a2 + 3a2z2 + a2z4 - 2a4z-2 - 3a4 - 2a4z2 - 3a4z4 - a4z6 + a6z-2 - 2a6 - 5a6z2 - 4a6z4 - a6z6 + 3a8 + 4a8z2 + a8z4 - a10

Kauffman Polynomial:

a2z-2 - 4a2 + 6a2z2 - 4a2z4 + a2z6 - 2a3z-1 + 4a3z - 5a3z5 + 2a3z7 + 2a4z-2 - 6a4 + 9a4z2 - 9a4z4 - 2a4z6 + 2a4z8 - 2a5z-1 + 2a5z + 7a5z3 - 12a5z5 + 2a5z7 + a5z9 + a6z-2 - a6 - 5a6z2 + 14a6z4 - 15a6z6 + 5a6z8 - 6a7z + 20a7z3 - 14a7z5 + 2a7z7 + a7z9 + 3a8 - 12a8z2 + 22a8z4 - 12a8z6 + 3a8z8 - 6a9z + 14a9z3 - 7a9z5 + 2a9z7 + a10 - 4a10z2 + 3a10z4 - 2a11z + a11z3

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                   1 j = -1                 1   j = -3               5 1   j = -5             3 3     j = -7           5 3       j = -9       1 4 3         j = -11       5 5           j = -13     2 5             j = -15   1 3               j = -17   2                 j = -19 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 374]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 374]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 9, 3, 10], X[12, 3, 13, 4], X[10, 5, 11, 6], > X[16, 11, 5, 12], X[4, 15, 1, 16], X[13, 20, 14, 21], X[7, 19, 8, 18], > X[17, 9, 18, 8], X[19, 22, 20, 17], X[21, 14, 22, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -6}, {-9, 8, -10, 7, -11, 10}, > {4, -1, -8, 9, 2, -4, 5, -3, -7, 11, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-9 3 5 8 8 8 6 6 2 1 - q + -- - -- + -- - -- + -- - -- + -- - - 8 7 6 5 4 3 2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -26 2 -22 2 2 4 2 6 3 4 2 1 - q + q + --- - q + --- + --- + --- + --- + --- + -- + -- + -- 24 20 16 14 12 10 8 6 4 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 374]][a, z]
Out[8]=	2 4 6 2 4 6 8 10 a 2 a a 2 2 4 2 3 a - 3 a - 2 a + 3 a - a + -- - ---- + -- + 3 a z - 2 a z - 2 2 2 z z z 6 2 8 2 2 4 4 4 6 4 8 4 4 6 6 6 > 5 a z + 4 a z + a z - 3 a z - 4 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 374]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 8 10 a 2 a a 2 a 2 a 3 -4 a - 6 a - a + 3 a + a + -- + ---- + -- - ---- - ---- + 4 a z + 2 2 2 z z z z z 5 7 9 11 2 2 4 2 6 2 > 2 a z - 6 a z - 6 a z - 2 a z + 6 a z + 9 a z - 5 a z - 8 2 10 2 5 3 7 3 9 3 11 3 2 4 > 12 a z - 4 a z + 7 a z + 20 a z + 14 a z + a z - 4 a z - 4 4 6 4 8 4 10 4 3 5 5 5 7 5 > 9 a z + 14 a z + 22 a z + 3 a z - 5 a z - 12 a z - 14 a z - 9 5 2 6 4 6 6 6 8 6 3 7 5 7 > 7 a z + a z - 2 a z - 15 a z - 12 a z + 2 a z + 2 a z + 7 7 9 7 4 8 6 8 8 8 5 9 7 9 > 2 a z + 2 a z + 2 a z + 5 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 2 1 3 2 5 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 19 7 17 6 15 6 15 5 13 5 13 4 11 4 q q q t q t q t q t q t q t q t 1 5 4 3 5 3 3 t t 2 > ----- + ------ + ----- + ----- + ----- + ---- + ---- + -- + - + q t 9 4 11 3 9 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n374

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