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

K11n124 K11n126

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X5,17,6,16 X7,19,8,18 X14,10,15,9 X2,11,3,12 X20,14,21,13 X22,15,1,16 X17,7,18,6 X12,20,13,19 X8,21,9,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-3 - 6t-2 + 15t-1 - 19 + 15t - 6t2 + t3

Conway Polynomial:

1 + z6

Other knots with the same Alexander/Conway Polynomial:

{K11n176, ...}

Determinant and Signature:

{63, 2}

Jones Polynomial:

q-1 - 3 + 7q - 9q2 + 11q3 - 11q4 + 9q5 - 7q6 + 4q7 - q8

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

{...}

A2 (sl(3)) Invariant:

q-4 - 1 + 3q2 - q4 + 2q6 + q8 - 2q10 + q12 - 3q14 + 2q16 - q20 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6z2 - a-6z4 + 3a-4z2 + 3a-4z4 + a-4z6 - 3a-2z2 - 2a-2z4 + 1 + z2

Kauffman Polynomial:

- a-9z3 + a-9z5 + a-8z2 - 7a-8z4 + 4a-8z6 + 4a-7z3 - 12a-7z5 + 6a-7z7 + a-6z2 - 3a-6z4 - 4a-6z6 + 4a-6z8 + 8a-5z3 - 14a-5z5 + 6a-5z7 + a-5z9 - 3a-4z2 + 9a-4z4 - 9a-4z6 + 5a-4z8 + 2a-3z5 + a-3z9 - 5a-2z2 + 6a-2z4 - a-2z6 + a-2z8 - 3a-1z3 + 3a-1z5 + 1 - 2z2 + z4

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

{0, 0}

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 11125. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                   1 j = 15                 3   j = 13               4 1   j = 11             5 3     j = 9           6 4       j = 7         5 5         j = 5       4 6           j = 3     3 5             j = 1   1 5               j = -1   2                 j = -3 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, 125]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 125]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[5, 17, 6, 16], X[7, 19, 8, 18], > X[14, 10, 15, 9], X[2, 11, 3, 12], X[20, 14, 21, 13], X[22, 15, 1, 16], > X[17, 7, 18, 6], X[12, 20, 13, 19], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 125]]
Out[4]=	GaussCode[1, -6, 2, -1, -3, 9, -4, -11, 5, -2, 6, -10, 7, -5, 8, 3, -9, 4, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, NonAlternating, 125]]
Out[5]=	DTCode[4, 10, -16, -18, 14, 2, 20, 22, -6, 12, 8]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 125]][t]
Out[6]=	-3 6 15 2 3 -19 + t - -- + -- + 15 t - 6 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 125]][z]
Out[7]=	6 1 + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 125], Knot[11, NonAlternating, 176]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 125]], KnotSignature[Knot[11, NonAlternating, 125]]}
Out[9]=	{63, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 125]][q]
Out[10]=	1 2 3 4 5 6 7 8 -3 + - + 7 q - 9 q + 11 q - 11 q + 9 q - 7 q + 4 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 125]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 125]][q]
Out[12]=	-4 2 4 6 8 10 12 14 16 20 22 -1 + q + 3 q - q + 2 q + q - 2 q + q - 3 q + 2 q - q + 2 q - 24 > q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 125]][a, z]
Out[13]=	2 2 2 4 4 4 6 2 z 3 z 3 z z 3 z 2 z z 1 + z - -- + ---- - ---- - -- + ---- - ---- + -- 6 4 2 6 4 2 4 a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 125]][a, z]
Out[14]=	2 2 2 2 3 3 3 3 4 4 2 z z 3 z 5 z z 4 z 8 z 3 z 4 7 z 3 z 1 - 2 z + -- + -- - ---- - ---- - -- + ---- + ---- - ---- + z - ---- - ---- + 8 6 4 2 9 7 5 a 8 6 a a a a a a a a a 4 4 5 5 5 5 5 6 6 6 6 9 z 6 z z 12 z 14 z 2 z 3 z 4 z 4 z 9 z z > ---- + ---- + -- - ----- - ----- + ---- + ---- + ---- - ---- - ---- - -- + 4 2 9 7 5 3 a 8 6 4 2 a a a a a a a a a a 7 7 8 8 8 9 9 6 z 6 z 4 z 5 z z z z > ---- + ---- + ---- + ---- + -- + -- + -- 7 5 6 4 2 5 3 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 125]], Vassiliev[3][Knot[11, NonAlternating, 125]]}
Out[15]=	{0, 0}
In[16]:=	Kh[Knot[11, NonAlternating, 125]][q, t]
Out[16]=	3 1 2 q 3 5 5 2 7 2 7 3 5 q + 3 q + ----- + --- + - + 5 q t + 4 q t + 6 q t + 5 q t + 5 q t + 3 2 q t t q t 9 3 9 4 11 4 11 5 13 5 13 6 15 6 > 6 q t + 4 q t + 5 q t + 3 q t + 4 q t + q t + 3 q t + 17 7 > q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n125

K11n124 K11n126