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

K11a13 K11a15

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X12,5,13,6 X2837 X14,9,15,10 X18,11,19,12 X6,13,7,14 X20,16,21,15 X10,17,11,18 X22,20,1,19 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 12 2 14 18 6 20 10 22 16

Alexander Polynomial:

t-4 - 5t-3 + 15t-2 - 28t-1 + 35 - 28t + 15t2 - 5t3 + t4

Conway Polynomial:

1 + 3z2 + 5z4 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{133, 0}

Jones Polynomial:

- q-5 + 3q-4 - 8q-3 + 14q-2 - 18q-1 + 22 - 21q + 19q2 - 14q3 + 8q4 - 4q5 + q6

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

{...}

A2 (sl(3)) Invariant:

- q-14 + q-12 - 4q-10 + q-8 + q-6 - 2q-4 + 7q-2 - 1 + 5q2 - 2q6 + 2q8 - 5q10 + q12 - q16 + q18

HOMFLY-PT Polynomial:

a-4 + 2a-4z2 + a-4z4 - 6a-2 - 12a-2z2 - 8a-2z4 - 2a-2z6 + 10 + 20z2 + 16z4 + 6z6 + z8 - 4a2 - 7a2z2 - 4a2z4 - a2z6

Kauffman Polynomial:

a-6z2 - 2a-6z4 + a-6z6 - 2a-5z + 8a-5z3 - 10a-5z5 + 4a-5z7 + a-4 - 2a-4z2 + 6a-4z4 - 12a-4z6 + 6a-4z8 - 10a-3z + 31a-3z3 - 32a-3z5 + 5a-3z7 + 4a-3z9 + 6a-2 - 21a-2z2 + 41a-2z4 - 46a-2z6 + 17a-2z8 + a-2z10 - 16a-1z + 43a-1z3 - 36a-1z5 - a-1z7 + 9a-1z9 + 10 - 32z2 + 54z4 - 50z6 + 19z8 + z10 - 12az + 29az3 - 24az5 + 4az7 + 5az9 + 4a2 - 13a2z2 + 17a2z4 - 14a2z6 + 8a2z8 - 3a3z + 7a3z3 - 9a3z5 + 6a3z7 + a4z2 - 4a4z4 + 3a4z6 + a5z - 2a5z3 + a5z5

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

{3, -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=0 is the signature of 1114. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     3   j = 9                   5 1   j = 7                 9 3     j = 5               10 5       j = 3             11 9         j = 1           11 10           j = -1         8 12             j = -3       6 10               j = -5     2 8                 j = -7   1 6                   j = -9   2                     j = -11 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, Alternating, 14]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 14]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 5, 13, 6], X[2, 8, 3, 7], > X[14, 9, 15, 10], X[18, 11, 19, 12], X[6, 13, 7, 14], X[20, 16, 21, 15], > X[10, 17, 11, 18], X[22, 20, 1, 19], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 14]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, 5, -9, 6, -3, 7, -5, 8, -11, 9, -6, 10, > -8, 11, -10]
In[5]:=	DTCode[Knot[11, Alternating, 14]]
Out[5]=	DTCode[4, 8, 12, 2, 14, 18, 6, 20, 10, 22, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 14]][t]
Out[6]=	-4 5 15 28 2 3 4 35 + t - -- + -- - -- - 28 t + 15 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 14]][z]
Out[7]=	2 4 6 8 1 + 3 z + 5 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 14]}
In[9]:=	{KnotDet[Knot[11, Alternating, 14]], KnotSignature[Knot[11, Alternating, 14]]}
Out[9]=	{133, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 14]][q]
Out[10]=	-5 3 8 14 18 2 3 4 5 6 22 - q + -- - -- + -- - -- - 21 q + 19 q - 14 q + 8 q - 4 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 14]}
In[12]:=	A2Invariant[Knot[11, Alternating, 14]][q]
Out[12]=	-14 -12 4 -8 -6 2 7 2 6 8 10 -1 - q + q - --- + q + q - -- + -- + 5 q - 2 q + 2 q - 5 q + 10 4 2 q q q 12 16 18 > q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 14]][a, z]
Out[13]=	2 2 4 4 -4 6 2 2 2 z 12 z 2 2 4 z 8 z 10 + a - -- - 4 a + 20 z + ---- - ----- - 7 a z + 16 z + -- - ---- - 2 4 2 4 2 a a a a a 6 2 4 6 2 z 2 6 8 > 4 a z + 6 z - ---- - a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 14]][a, z]
Out[14]=	-4 6 2 2 z 10 z 16 z 3 5 2 10 + a + -- + 4 a - --- - ---- - ---- - 12 a z - 3 a z + a z - 32 z + 2 5 3 a a a a 2 2 2 3 3 3 z 2 z 21 z 2 2 4 2 8 z 31 z 43 z 3 > -- - ---- - ----- - 13 a z + a z + ---- + ----- + ----- + 29 a z + 6 4 2 5 3 a a a a a a 4 4 4 3 3 5 3 4 2 z 6 z 41 z 2 4 4 4 > 7 a z - 2 a z + 54 z - ---- + ---- + ----- + 17 a z - 4 a z - 6 4 2 a a a 5 5 5 6 6 10 z 32 z 36 z 5 3 5 5 5 6 z 12 z > ----- - ----- - ----- - 24 a z - 9 a z + a z - 50 z + -- - ----- - 5 3 a 6 4 a a a a 6 7 7 7 46 z 2 6 4 6 4 z 5 z z 7 3 7 8 > ----- - 14 a z + 3 a z + ---- + ---- - -- + 4 a z + 6 a z + 19 z + 2 5 3 a a a a 8 8 9 9 10 6 z 17 z 2 8 4 z 9 z 9 10 z > ---- + ----- + 8 a z + ---- + ---- + 5 a z + z + --- 4 2 3 a 2 a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 14]], Vassiliev[3][Knot[11, Alternating, 14]]}
Out[15]=	{3, -1}
In[16]:=	Kh[Knot[11, Alternating, 14]][q, t]
Out[16]=	12 1 2 1 6 2 8 6 10 -- + 11 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 8 3 3 2 5 2 5 3 7 3 7 4 > --- + 10 q t + 11 q t + 9 q t + 10 q t + 5 q t + 9 q t + 3 q t + q t 9 4 9 5 11 5 13 6 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a14

K11a13 K11a15