© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11n130 K11n132

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X18,5,19,6 X7,16,8,17 X9,14,10,15 X2,11,3,12 X20,14,21,13 X15,8,16,9 X22,17,1,18 X12,20,13,19 X6,21,7,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 18 -16 -14 2 20 -8 22 12 6

Alexander Polynomial:

t-3 - 6t-2 + 16t-1 - 21 + 16t - 6t2 + t3

Conway Polynomial:

1 + z2 + z6

Other knots with the same Alexander/Conway Polynomial:

{K11n7, K11n160, ...}

Determinant and Signature:

{67, -2}

Jones Polynomial:

q-7 - 4q-6 + 7q-5 - 10q-4 + 12q-3 - 11q-2 + 10q-1 - 7 + 4q - q2

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

q-22 - q-20 - 2q-18 + 2q-16 - 2q-14 + q-12 + q-10 - q-8 + 3q-6 - 2q-4 + 3q-2 - q2 + 2q4 - q6

HOMFLY-PT Polynomial:

- z2 - z4 + 2a2 + 4a2z2 + 3a2z4 + a2z6 - a4 - 3a4z2 - 2a4z4 + a6z2

Kauffman Polynomial:

- a-1z3 + a-1z5 + 2z2 - 7z4 + 4z6 + 3az3 - 11az5 + 6az7 - 2a2 + 6a2z2 - 8a2z4 - 2a2z6 + 4a2z8 + a3z + 5a3z3 - 14a3z5 + 7a3z7 + a3z9 - a4 + 4a4z2 - a4z4 - 5a4z6 + 5a4z8 + a5z - 3a5z3 + 2a5z5 + a5z7 + a5z9 - a6z2 + a6z4 + a6z6 + a6z8 - 4a7z3 + 4a7z5 - a8z2 + a8z4

V2 and V3, the type 2 and 3 Vassiliev invariants:

{1, -1}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=-2 is the signature of 11131. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 5                   1 j = 3                 3   j = 1               4 1   j = -1             6 3     j = -3           6 5       j = -5         6 5         j = -7       4 6           j = -9     3 6             j = -11   1 4               j = -13   3                 j = -15 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]:=	Show[DrawMorseLink[Knot[11, NonAlternating, 131]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 131]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[18, 5, 19, 6], X[7, 16, 8, 17], > X[9, 14, 10, 15], X[2, 11, 3, 12], X[20, 14, 21, 13], X[15, 8, 16, 9], > X[22, 17, 1, 18], X[12, 20, 13, 19], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 131]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -11, -4, 8, -5, -2, 6, -10, 7, 5, -8, 4, 9, -3, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 131]]
Out[5]=	DTCode[4, 10, 18, -16, -14, 2, 20, -8, 22, 12, 6]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 131]][t]
Out[6]=	-3 6 16 2 3 -21 + t - -- + -- + 16 t - 6 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 131]][z]
Out[7]=	2 6 1 + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 7], Knot[11, NonAlternating, 131], > Knot[11, NonAlternating, 160]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 131]], KnotSignature[Knot[11, NonAlternating, 131]]}
Out[9]=	{67, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 131]][q]
Out[10]=	-7 4 7 10 12 11 10 2 -7 + q - -- + -- - -- + -- - -- + -- + 4 q - q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 131]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 131]][q]
Out[12]=	-22 -20 2 2 2 -12 -10 -8 3 2 3 2 4 q - q - --- + --- - --- + q + q - q + -- - -- + -- - q + 2 q - 18 16 14 6 4 2 q q q q q q 6 > q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 131]][a, z]
Out[13]=	2 4 2 2 2 4 2 6 2 4 2 4 4 4 2 6 2 a - a - z + 4 a z - 3 a z + a z - z + 3 a z - 2 a z + a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 131]][a, z]
Out[14]=	3 2 4 3 5 2 2 2 4 2 6 2 8 2 z -2 a - a + a z + a z + 2 z + 6 a z + 4 a z - a z - a z - -- + a 3 3 3 5 3 7 3 4 2 4 4 4 6 4 > 3 a z + 5 a z - 3 a z - 4 a z - 7 z - 8 a z - a z + a z + 5 8 4 z 5 3 5 5 5 7 5 6 2 6 > a z + -- - 11 a z - 14 a z + 2 a z + 4 a z + 4 z - 2 a z - a 4 6 6 6 7 3 7 5 7 2 8 4 8 6 8 > 5 a z + a z + 6 a z + 7 a z + a z + 4 a z + 5 a z + a z + 3 9 5 9 > a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 131]], Vassiliev[3][Knot[11, NonAlternating, 131]]}
Out[15]=	{1, -1}
In[16]:=	Kh[Knot[11, NonAlternating, 131]][q, t]
Out[16]=	5 6 1 3 1 4 3 6 4 6 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 6 5 6 3 t 2 3 2 5 3 > ----- + ---- + ---- + --- + 4 q t + q t + 3 q t + q t 5 2 5 3 q q t q t q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n131

K11n130 K11n132