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

K11a15 K11a17

Knotscape

This page is passe. Go here instead!    The Knot K11a16 Visit K11a16's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 12 2 14 18 6 22 20 10 16

Alexander Polynomial:

- 2t-3 + 10t-2 - 24t-1 + 33 - 24t + 10t2 - 2t3

Conway Polynomial:

1 - 2z2 - 2z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{105, 0}

Jones Polynomial:

q-6 - 3q-5 + 7q-4 - 11q-3 + 14q-2 - 17q-1 + 17 - 14q + 11q2 - 6q3 + 3q4 - q5

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

{K11a280, ...}

A2 (sl(3)) Invariant:

q-18 - q-16 + 2q-14 + 2q-12 - 3q-10 + 2q-8 - 3q-6 - 2q-4 + q-2 - 2 + 4q2 - q4 + 2q6 + 3q8 - 2q10 + q12 - q16

HOMFLY-PT Polynomial:

- a-4 - a-4z2 + 3a-2 + 4a-2z2 + 2a-2z4 - 2z2 - 2z4 - z6 - 3a2 - 5a2z2 - 3a2z4 - a2z6 + 2a4 + 2a4z2 + a4z4

Kauffman Polynomial:

a-5z - 2a-5z3 + a-5z5 - a-4 + 4a-4z2 - 6a-4z4 + 3a-4z6 + 2a-3z - 2a-3z3 - 4a-3z5 + 4a-3z7 - 3a-2 + 7a-2z2 - 7a-2z4 - a-2z6 + 4a-2z8 + 3a-1z - a-1z3 - 4a-1z5 + a-1z7 + 3a-1z9 - 7z2 + 20z4 - 20z6 + 8z8 + z10 + 5az - 8az3 + 11az5 - 13az7 + 7az9 + 3a2 - 20a2z2 + 37a2z4 - 31a2z6 + 9a2z8 + a2z10 + 2a3z - 2a3z3 + 2a3z5 - 7a3z7 + 4a3z9 + 2a4 - 8a4z2 + 13a4z4 - 14a4z6 + 5a4z8 - a5z + 5a5z3 - 8a5z5 + 3a5z7 + 2a6z2 - 3a6z4 + a6z6

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

{-2, 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=0 is the signature of 1116. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     2   j = 7                   4 1   j = 5                 7 2     j = 3               7 4       j = 1             10 7         j = -1           8 8           j = -3         6 9             j = -5       5 8               j = -7     2 6                 j = -9   1 5                   j = -11   2                     j = -13 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, 16]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 16]]
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, 12, 19, 11], X[6, 13, 7, 14], X[22, 15, 1, 16], > X[20, 17, 21, 18], X[10, 20, 11, 19], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 16]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -3, 7, -5, 8, -11, 9, -6, 10, > -9, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 16]]
Out[5]=	DTCode[4, 8, 12, 2, 14, 18, 6, 22, 20, 10, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 16]][t]
Out[6]=	2 10 24 2 3 33 - -- + -- - -- - 24 t + 10 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 16]][z]
Out[7]=	2 4 6 1 - 2 z - 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 16]}
In[9]:=	{KnotDet[Knot[11, Alternating, 16]], KnotSignature[Knot[11, Alternating, 16]]}
Out[9]=	{105, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 16]][q]
Out[10]=	-6 3 7 11 14 17 2 3 4 5 17 + q - -- + -- - -- + -- - -- - 14 q + 11 q - 6 q + 3 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 16], Knot[11, Alternating, 280]}
In[12]:=	A2Invariant[Knot[11, Alternating, 16]][q]
Out[12]=	-18 -16 2 2 3 2 3 2 -2 2 4 6 -2 + q - q + --- + --- - --- + -- - -- - -- + q + 4 q - q + 2 q + 14 12 10 8 6 4 q q q q q q 8 10 12 16 > 3 q - 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 16]][a, z]
Out[13]=	2 2 4 -4 3 2 4 2 z 4 z 2 2 4 2 4 2 z -a + -- - 3 a + 2 a - 2 z - -- + ---- - 5 a z + 2 a z - 2 z + ---- - 2 4 2 2 a a a a 2 4 4 4 6 2 6 > 3 a z + a z - z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 16]][a, z]
Out[14]=	-4 3 2 4 z 2 z 3 z 3 5 2 -a - -- + 3 a + 2 a + -- + --- + --- + 5 a z + 2 a z - a z - 7 z + 2 5 3 a a a a 2 2 3 3 3 4 z 7 z 2 2 4 2 6 2 2 z 2 z z 3 > ---- + ---- - 20 a z - 8 a z + 2 a z - ---- - ---- - -- - 8 a z - 4 2 5 3 a a a a a 4 4 3 3 5 3 4 6 z 7 z 2 4 4 4 6 4 > 2 a z + 5 a z + 20 z - ---- - ---- + 37 a z + 13 a z - 3 a z + 4 2 a a 5 5 5 6 6 z 4 z 4 z 5 3 5 5 5 6 3 z z > -- - ---- - ---- + 11 a z + 2 a z - 8 a z - 20 z + ---- - -- - 5 3 a 4 2 a a a a 7 7 2 6 4 6 6 6 4 z z 7 3 7 5 7 > 31 a z - 14 a z + a z + ---- + -- - 13 a z - 7 a z + 3 a z + 3 a a 8 9 8 4 z 2 8 4 8 3 z 9 3 9 10 2 10 > 8 z + ---- + 9 a z + 5 a z + ---- + 7 a z + 4 a z + z + a z 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 16]], Vassiliev[3][Knot[11, Alternating, 16]]}
Out[15]=	{-2, 3}
In[16]:=	Kh[Knot[11, Alternating, 16]][q, t]
Out[16]=	8 1 2 1 5 2 6 5 8 - + 10 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 6 9 8 3 3 2 5 2 5 3 > ----- + ---- + --- + 7 q t + 7 q t + 4 q t + 7 q t + 2 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a16

K11a15 K11a17