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

K11n24 K11n26

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X12,5,13,6 X2837 X9,15,10,14 X11,18,12,19 X6,13,7,14 X15,21,16,20 X17,1,18,22 X19,10,20,11 X21,17,22,16

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-3 - 5t-2 + 11t-1 - 13 + 11t - 5t2 + t3

Conway Polynomial:

1 + z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{926, ...}

Determinant and Signature:

{47, 2}

Jones Polynomial:

q-1 - 2 + 5q - 7q2 + 8q3 - 8q4 + 7q5 - 5q6 + 3q7 - q8

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

{925, ...}

A2 (sl(3)) Invariant:

q-4 + q-2 + 2q2 - 2q4 - q10 + 2q12 - q14 + 2q16 - q20 + q22 - q24

HOMFLY-PT Polynomial:

- a-6 - 2a-6z2 - a-6z4 + 3a-4 + 6a-4z2 + 4a-4z4 + a-4z6 - 3a-2 - 5a-2z2 - 2a-2z4 + 2 + z2

Kauffman Polynomial:

- 2a-9z3 + a-9z5 + 2a-8z2 - 7a-8z4 + 3a-8z6 - 2a-7z + 6a-7z3 - 10a-7z5 + 4a-7z7 + a-6 - 3a-6z2 + 6a-6z4 - 7a-6z6 + 3a-6z8 - 4a-5z + 11a-5z3 - 7a-5z5 + a-5z7 + a-5z9 + 3a-4 - 14a-4z2 + 24a-4z4 - 14a-4z6 + 4a-4z8 - 2a-3z + 6a-3z5 - 3a-3z7 + a-3z9 + 3a-2 - 12a-2z2 + 12a-2z4 - 4a-2z6 + a-2z8 - 3a-1z3 + 2a-1z5 + 2 - 3z2 + z4

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

{0, 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 1125. 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                 2   j = 13               3 1   j = 11             4 2     j = 9           4 3       j = 7         4 4         j = 5       3 4           j = 3     2 4             j = 1   1 4               j = -1   1                 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, 25]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 25]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 5, 13, 6], X[2, 8, 3, 7], > X[9, 15, 10, 14], X[11, 18, 12, 19], X[6, 13, 7, 14], X[15, 21, 16, 20], > X[17, 1, 18, 22], X[19, 10, 20, 11], X[21, 17, 22, 16]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 25]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, -5, 10, -6, -3, 7, 5, -8, 11, -9, 6, -10, > 8, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 25]]
Out[5]=	DTCode[4, 8, 12, 2, -14, -18, 6, -20, -22, -10, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 25]][t]
Out[6]=	-3 5 11 2 3 -13 + t - -- + -- + 11 t - 5 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 25]][z]
Out[7]=	4 6 1 + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 26], Knot[11, NonAlternating, 25]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 25]], KnotSignature[Knot[11, NonAlternating, 25]]}
Out[9]=	{47, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 25]][q]
Out[10]=	1 2 3 4 5 6 7 8 -2 + - + 5 q - 7 q + 8 q - 8 q + 7 q - 5 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 25], Knot[11, NonAlternating, 25]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 25]][q]
Out[12]=	-4 -2 2 4 10 12 14 16 20 22 24 q + q + 2 q - 2 q - q + 2 q - q + 2 q - q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 25]][a, z]
Out[13]=	2 2 2 4 4 4 6 -6 3 3 2 2 z 6 z 5 z z 4 z 2 z z 2 - a + -- - -- + z - ---- + ---- - ---- - -- + ---- - ---- + -- 4 2 6 4 2 6 4 2 4 a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 25]][a, z]
Out[14]=	2 2 2 2 -6 3 3 2 z 4 z 2 z 2 2 z 3 z 14 z 12 z 2 + a + -- + -- - --- - --- - --- - 3 z + ---- - ---- - ----- - ----- - 4 2 7 5 3 8 6 4 2 a a a a a a a a a 3 3 3 3 4 4 4 4 5 2 z 6 z 11 z 3 z 4 7 z 6 z 24 z 12 z z > ---- + ---- + ----- - ---- + z - ---- + ---- + ----- + ----- + -- - 9 7 5 a 8 6 4 2 9 a a a a a a a a 5 5 5 5 6 6 6 6 7 7 10 z 7 z 6 z 2 z 3 z 7 z 14 z 4 z 4 z z > ----- - ---- + ---- + ---- + ---- - ---- - ----- - ---- + ---- + -- - 7 5 3 a 8 6 4 2 7 5 a a a a a a a a a 7 8 8 8 9 9 3 z 3 z 4 z z z z > ---- + ---- + ---- + -- + -- + -- 3 6 4 2 5 3 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 25]], Vassiliev[3][Knot[11, NonAlternating, 25]]}
Out[15]=	{0, 1}
In[16]:=	Kh[Knot[11, NonAlternating, 25]][q, t]
Out[16]=	3 1 1 q 3 5 5 2 7 2 7 3 4 q + 2 q + ----- + --- + - + 4 q t + 3 q t + 4 q t + 4 q t + 4 q t + 3 2 q t t q t 9 3 9 4 11 4 11 5 13 5 13 6 15 6 > 4 q t + 3 q t + 4 q t + 2 q t + 3 q t + q t + 2 q t + 17 7 > q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n25

K11n24 K11n26