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

K11n125 K11n127

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X5,19,6,18 X7,13,8,12 X2,10,3,9 X11,17,12,16 X13,21,14,20 X15,9,16,8 X17,1,18,22 X19,15,20,14 X21,7,22,6

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

3t-3 - 6t-2 + 4t-1 - 1 + 4t - 6t2 + 3t3

Conway Polynomial:

1 + 7z2 + 12z4 + 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{27, 6}

Jones Polynomial:

q3 - 2q4 + 4q5 - 3q6 + 5q7 - 5q8 + 3q9 - 3q10 + q11

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

{...}

A2 (sl(3)) Invariant:

q10 - q12 + q14 + q16 + 3q18 + 2q20 + q22 + 2q24 - 2q26 - q28 - 3q30 - 2q32 - q34 + 2q38

HOMFLY-PT Polynomial:

2a-12 - 7a-10 - 9a-10z2 - 2a-10z4 + 5a-8 + 13a-8z2 + 10a-8z4 + 2a-8z6 + a-6 + 3a-6z2 + 4a-6z4 + a-6z6

Kauffman Polynomial:

- 3a-13z + 2a-13z3 + 2a-12 - 3a-12z2 + 8a-12z4 - 5a-12z6 + a-12z8 - 13a-11z + 19a-11z3 - 4a-11z5 - 3a-11z7 + a-11z9 + 7a-10 - 19a-10z2 + 31a-10z4 - 20a-10z6 + 4a-10z8 - 10a-9z + 20a-9z3 - 11a-9z5 - a-9z7 + a-9z9 + 5a-8 - 13a-8z2 + 19a-8z4 - 14a-8z6 + 3a-8z8 + 3a-7z3 - 7a-7z5 + 2a-7z7 - a-6 + 3a-6z2 - 4a-6z4 + a-6z6

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

{7, 16}

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=6 is the signature of 11126. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 23                 1 j = 21               2   j = 19             1 1   j = 17           4 2     j = 15         2 2       j = 13       2 4         j = 11     2 2 1         j = 9     2             j = 7 1 2               j = 5 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, 126]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 126]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[5, 19, 6, 18], X[7, 13, 8, 12], > X[2, 10, 3, 9], X[11, 17, 12, 16], X[13, 21, 14, 20], X[15, 9, 16, 8], > X[17, 1, 18, 22], X[19, 15, 20, 14], X[21, 7, 22, 6]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 126]]
Out[4]=	GaussCode[1, -5, 2, -1, -3, 11, -4, 8, 5, -2, -6, 4, -7, 10, -8, 6, -9, 3, -10, > 7, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 126]]
Out[5]=	DTCode[4, 10, -18, -12, 2, -16, -20, -8, -22, -14, -6]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 126]][t]
Out[6]=	3 6 4 2 3 -1 + -- - -- + - + 4 t - 6 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 126]][z]
Out[7]=	2 4 6 1 + 7 z + 12 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 126]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 126]], KnotSignature[Knot[11, NonAlternating, 126]]}
Out[9]=	{27, 6}
In[10]:=	J=Jones[Knot[11, NonAlternating, 126]][q]
Out[10]=	3 4 5 6 7 8 9 10 11 q - 2 q + 4 q - 3 q + 5 q - 5 q + 3 q - 3 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 126]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 126]][q]
Out[12]=	10 12 14 16 18 20 22 24 26 28 30 q - q + q + q + 3 q + 2 q + q + 2 q - 2 q - q - 3 q - 32 34 38 > 2 q - q + 2 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 126]][a, z]
Out[13]=	2 2 2 4 4 4 6 6 2 7 5 -6 9 z 13 z 3 z 2 z 10 z 4 z 2 z z --- - --- + -- + a - ---- + ----- + ---- - ---- + ----- + ---- + ---- + -- 12 10 8 10 8 6 10 8 6 8 6 a a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 126]][a, z]
Out[14]=	2 2 2 2 3 2 7 5 -6 3 z 13 z 10 z 3 z 19 z 13 z 3 z 2 z --- + --- + -- - a - --- - ---- - ---- - ---- - ----- - ----- + ---- + ---- + 12 10 8 13 11 9 12 10 8 6 13 a a a a a a a a a a a 3 3 3 4 4 4 4 5 5 5 19 z 20 z 3 z 8 z 31 z 19 z 4 z 4 z 11 z 7 z > ----- + ----- + ---- + ---- + ----- + ----- - ---- - ---- - ----- - ---- - 11 9 7 12 10 8 6 11 9 7 a a a a a a a a a a 6 6 6 6 7 7 7 8 8 8 9 9 5 z 20 z 14 z z 3 z z 2 z z 4 z 3 z z z > ---- - ----- - ----- + -- - ---- - -- + ---- + --- + ---- + ---- + --- + -- 12 10 8 6 11 9 7 12 10 8 11 9 a a a a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 126]], Vassiliev[3][Knot[11, NonAlternating, 126]]}
Out[15]=	{7, 16}
In[16]:=	Kh[Knot[11, NonAlternating, 126]][q, t]
Out[16]=	5 7 7 9 2 11 2 11 3 13 3 11 4 q + q + 2 q t + 2 q t + 2 q t + 2 q t + 2 q t + q t + 13 4 15 4 15 5 17 5 17 6 19 6 19 7 > 4 q t + 2 q t + 2 q t + 4 q t + 2 q t + q t + q t + 21 7 23 8 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n126

K11n125 K11n127