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

K11n14 K11n16

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X7,16,8,17 X2,9,3,10 X11,19,12,18 X13,21,14,20 X15,6,16,7 X17,1,18,22 X19,13,20,12 X21,15,22,14

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-3 - 4t-2 + 8t-1 - 9 + 8t - 4t2 + t3

Conway Polynomial:

1 + z2 + 2z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{816, 10156, K11n56, K11n58, ...}

Determinant and Signature:

{35, 2}

Jones Polynomial:

q-1 - 2 + 4q - 5q2 + 6q3 - 6q4 + 5q5 - 3q6 + 2q7 - q8

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

{912, ...}

A2 (sl(3)) Invariant:

q-4 + q-2 + q2 - 2q4 + 2q12 + 2q16 - q20 - q24

HOMFLY-PT Polynomial:

- 2a-6 - 3a-6z2 - a-6z4 + 5a-4 + 9a-4z2 + 5a-4z4 + a-4z6 - 4a-2 - 6a-2z2 - 2a-2z4 + 2 + z2

Kauffman Polynomial:

a-9z - 3a-9z3 + a-9z5 + 3a-8z2 - 6a-8z4 + 2a-8z6 + 2a-7z3 - 5a-7z5 + 2a-7z7 + 2a-6 - 6a-6z2 + 10a-6z4 - 7a-6z6 + 2a-6z8 - a-5z + a-5z3 + 4a-5z5 - 3a-5z7 + a-5z9 + 5a-4 - 21a-4z2 + 29a-4z4 - 14a-4z6 + 3a-4z8 + a-3z - 8a-3z3 + 12a-3z5 - 5a-3z7 + a-3z9 + 4a-2 - 15a-2z2 + 14a-2z4 - 5a-2z6 + a-2z8 + a-1z - 4a-1z3 + 2a-1z5 + 2 - 3z2 + z4

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

{1, 3}

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 1115. 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                 1   j = 13               2 1   j = 11             3 1     j = 9           3 2       j = 7         3 3         j = 5       2 3           j = 3     2 3             j = 1   1 3               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, 15]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 15]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[7, 16, 8, 17], > X[2, 9, 3, 10], X[11, 19, 12, 18], X[13, 21, 14, 20], X[15, 6, 16, 7], > X[17, 1, 18, 22], X[19, 13, 20, 12], X[21, 15, 22, 14]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 15]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, 8, -4, -2, 5, -3, -6, 10, -7, 11, -8, 4, -9, 6, -10, > 7, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 15]]
Out[5]=	DTCode[4, 8, 10, -16, 2, -18, -20, -6, -22, -12, -14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 15]][t]
Out[6]=	-3 4 8 2 3 -9 + t - -- + - + 8 t - 4 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 15]][z]
Out[7]=	2 4 6 1 + z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 16], Knot[10, 156], Knot[11, NonAlternating, 15], > Knot[11, NonAlternating, 56], Knot[11, NonAlternating, 58]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 15]], KnotSignature[Knot[11, NonAlternating, 15]]}
Out[9]=	{35, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 15]][q]
Out[10]=	1 2 3 4 5 6 7 8 -2 + - + 4 q - 5 q + 6 q - 6 q + 5 q - 3 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 12], Knot[11, NonAlternating, 15]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 15]][q]
Out[12]=	-4 -2 2 4 12 16 20 24 q + q + q - 2 q + 2 q + 2 q - q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 15]][a, z]
Out[13]=	2 2 2 4 4 4 6 2 5 4 2 3 z 9 z 6 z z 5 z 2 z z 2 - -- + -- - -- + z - ---- + ---- - ---- - -- + ---- - ---- + -- 6 4 2 6 4 2 6 4 2 4 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 15]][a, z]
Out[14]=	2 2 2 2 2 5 4 z z z z 2 3 z 6 z 21 z 15 z 2 + -- + -- + -- + -- - -- + -- + - - 3 z + ---- - ---- - ----- - ----- - 6 4 2 9 5 3 a 8 6 4 2 a a a a a a a a a a 3 3 3 3 3 4 4 4 4 5 3 z 2 z z 8 z 4 z 4 6 z 10 z 29 z 14 z z > ---- + ---- + -- - ---- - ---- + z - ---- + ----- + ----- + ----- + -- - 9 7 5 3 a 8 6 4 2 9 a a a a a a a a a 5 5 5 5 6 6 6 6 7 7 5 z 4 z 12 z 2 z 2 z 7 z 14 z 5 z 2 z 3 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 5 z 2 z 3 z z z z > ---- + ---- + ---- + -- + -- + -- 3 6 4 2 5 3 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 15]], Vassiliev[3][Knot[11, NonAlternating, 15]]}
Out[15]=	{1, 3}
In[16]:=	Kh[Knot[11, NonAlternating, 15]][q, t]
Out[16]=	3 1 1 q 3 5 5 2 7 2 7 3 3 q + 2 q + ----- + --- + - + 3 q t + 2 q t + 3 q t + 3 q t + 3 q t + 3 2 q t t q t 9 3 9 4 11 4 11 5 13 5 13 6 15 6 17 7 > 3 q t + 2 q t + 3 q t + q t + 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n15

K11n14 K11n16