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L11n313 L11n315

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X3,12,4,13 X7,17,8,16 X9,20,10,21 X11,18,12,19 X19,22,20,11 X15,9,16,8 X21,10,22,5 X17,14,18,15 X2536 X13,4,14,1

Gauss Code:

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

Jones Polynomial:

- 2q-9 + 5q-8 - 7q-7 + 10q-6 - 10q-5 + 10q-4 - 7q-3 + 6q-2 - 2q-1 + 1

A2 (sl(3)) Invariant:

- q-32 - q-28 + q-26 + 2q-24 - q-22 + 3q-20 + 3q-16 + 4q-14 + 2q-12 + 6q-10 + 2q-8 + 4q-6 + 2q-4 + 1

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a2z-2 - 4a2 + 6a2z2 - 4a2z4 + a2z6 - 2a3z-1 + 4a3z + a3z3 - 5a3z5 + 2a3z7 + 2a4z-2 - 7a4 + 14a4z2 - 11a4z4 - a4z6 + 2a4z8 - 2a5z-1 + 6a5z - 10a5z5 + 3a5z7 + a5z9 + a6z-2 - 2a6 + a6z2 - 8a6z6 + 5a6z8 + 4a7z3 - 9a7z5 + 4a7z7 + a7z9 + 4a8 - 12a8z2 + 11a8z4 - 5a8z6 + 3a8z8 - 4a9z + 8a9z3 - 4a9z5 + 3a9z7 + 2a10 - 5a10z2 + 4a10z4 + a10z6 - 2a11z + 3a11z3

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 2     j = -7           7 4       j = -9         5 5         j = -11       5 5           j = -13     3 6             j = -15   2 4               j = -17   3                 j = -19 2

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, 314]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 314]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 12, 4, 13], X[7, 17, 8, 16], X[9, 20, 10, 21], > X[11, 18, 12, 19], X[19, 22, 20, 11], X[15, 9, 16, 8], X[21, 10, 22, 5], > X[17, 14, 18, 15], X[2, 5, 3, 6], X[13, 4, 14, 1]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -3, 7, -4, 8}, > {-5, 2, -11, 9, -7, 3, -9, 5, -6, 4, -8, 6}]
In[6]:=	Jones[L][q]
Out[6]=	2 5 7 10 10 10 7 6 2 1 - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -28 -26 2 -22 3 3 4 2 6 2 4 2 1 - q - 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, 314]][a, z]
Out[8]=	2 4 6 2 4 8 10 a 2 a a 2 2 4 2 6 2 3 a - 4 a + 2 a - a + -- - ---- + -- + 3 a z - 3 a z - 2 a z + 2 2 2 z z z 8 2 2 4 4 4 6 4 8 4 4 6 6 6 > 3 a z + a z - 3 a z - 3 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 314]][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 - 7 a - 2 a + 4 a + 2 a + -- + ---- + -- - ---- - ---- + 4 a z + 2 2 2 z z z z z 5 9 11 2 2 4 2 6 2 8 2 > 6 a z - 4 a z - 2 a z + 6 a z + 14 a z + a z - 12 a z - 10 2 3 3 7 3 9 3 11 3 2 4 4 4 > 5 a z + a z + 4 a z + 8 a z + 3 a z - 4 a z - 11 a z + 8 4 10 4 3 5 5 5 7 5 9 5 2 6 > 11 a z + 4 a z - 5 a z - 10 a z - 9 a z - 4 a z + a z - 4 6 6 6 8 6 10 6 3 7 5 7 7 7 > a z - 8 a z - 5 a z + a z + 2 a z + 3 a z + 4 a z + 9 7 4 8 6 8 8 8 5 9 7 9 > 3 a z + 2 a z + 5 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 5 2 3 2 4 3 6 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 5 5 5 7 4 3 t t 2 > ------ + ----- + ----- + ----- + ---- + ---- + -- + - + q t 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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n314

L11n313 L11n315