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

K11a17 K11a19

Knotscape

This page is passe. Go here instead!    The Knot K11a18 Visit K11a18's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 12 2 16 6 20 18 10 22 14

Alexander Polynomial:

3t-3 - 13t-2 + 29t-1 - 37 + 29t - 13t2 + 3t3

Conway Polynomial:

1 + 4z2 + 5z4 + 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{127, 2}

Jones Polynomial:

- q-2 + 3q-1 - 7 + 13q - 17q2 + 21q3 - 20q4 + 18q5 - 14q6 + 8q7 - 4q8 + q9

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

{...}

A2 (sl(3)) Invariant:

- q-6 + q-4 - q-2 - 2 + 4q2 - 3q4 + 3q6 + 3q8 + 5q12 - 3q14 + 2q16 - 2q18 - 4q20 + 2q22 - 2q24 + q28

HOMFLY-PT Polynomial:

a-8 + a-8z2 - 5a-6 - 7a-6z2 - 3a-6z4 + 5a-4 + 10a-4z2 + 7a-4z4 + 2a-4z6 + a-2 + 2a-2z2 + 2a-2z4 + a-2z6 - 1 - 2z2 - z4

Kauffman Polynomial:

a-10z2 - 2a-10z4 + a-10z6 - 2a-9z + 8a-9z3 - 10a-9z5 + 4a-9z7 + a-8 - a-8z2 + 5a-8z4 - 12a-8z6 + 6a-8z8 - 11a-7z + 30a-7z3 - 30a-7z5 + 4a-7z7 + 4a-7z9 + 5a-6 - 15a-6z2 + 31a-6z4 - 39a-6z6 + 15a-6z8 + a-6z10 - 14a-5z + 38a-5z3 - 33a-5z5 + 8a-5z9 + 5a-4 - 15a-4z2 + 31a-4z4 - 35a-4z6 + 15a-4z8 + a-4z10 - 6a-3z + 20a-3z3 - 20a-3z5 + 5a-3z7 + 4a-3z9 - a-2 + a-2z2 + 2a-2z4 - 6a-2z6 + 6a-2z8 + 2a-1z3 - 6a-1z5 + 5a-1z7 - 1 + 3z2 - 5z4 + 3z6 + az - 2az3 + az5

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

{4, 5}

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 1118. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                       1 j = 17                     3   j = 15                   5 1   j = 13                 9 3     j = 11               9 5       j = 9             11 9         j = 7           10 9           j = 5         7 11             j = 3       6 10               j = 1     2 8                 j = -1   1 5                   j = -3   2                     j = -5 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, 18]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 18]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 6, 13, 5], X[2, 8, 3, 7], > X[16, 10, 17, 9], X[6, 12, 7, 11], X[20, 13, 21, 14], X[18, 16, 19, 15], > X[10, 18, 11, 17], X[22, 19, 1, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 18]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -6, 4, -2, 5, -9, 6, -3, 7, -11, 8, -5, 9, -8, 10, > -7, 11, -10]
In[5]:=	DTCode[Knot[11, Alternating, 18]]
Out[5]=	DTCode[4, 8, 12, 2, 16, 6, 20, 18, 10, 22, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 18]][t]
Out[6]=	3 13 29 2 3 -37 + -- - -- + -- + 29 t - 13 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 18]][z]
Out[7]=	2 4 6 1 + 4 z + 5 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 18]}
In[9]:=	{KnotDet[Knot[11, Alternating, 18]], KnotSignature[Knot[11, Alternating, 18]]}
Out[9]=	{127, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 18]][q]
Out[10]=	-2 3 2 3 4 5 6 7 8 9 -7 - q + - + 13 q - 17 q + 21 q - 20 q + 18 q - 14 q + 8 q - 4 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 18]}
In[12]:=	A2Invariant[Knot[11, Alternating, 18]][q]
Out[12]=	-6 -4 -2 2 4 6 8 12 14 16 -2 - q + q - q + 4 q - 3 q + 3 q + 3 q + 5 q - 3 q + 2 q - 18 20 22 24 28 > 2 q - 4 q + 2 q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 18]][a, z]
Out[13]=	2 2 2 2 4 4 -8 5 5 -2 2 z 7 z 10 z 2 z 4 3 z 7 z -1 + a - -- + -- + a - 2 z + -- - ---- + ----- + ---- - z - ---- + ---- + 6 4 8 6 4 2 6 4 a a a a a a a a 4 6 6 2 z 2 z z > ---- + ---- + -- 2 4 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 18]][a, z]
Out[14]=	2 2 -8 5 5 -2 2 z 11 z 14 z 6 z 2 z z -1 + a + -- + -- - a - --- - ---- - ---- - --- + a z + 3 z + --- - -- - 6 4 9 7 5 3 10 8 a a a a a a a a 2 2 2 3 3 3 3 3 15 z 15 z z 8 z 30 z 38 z 20 z 2 z 3 4 > ----- - ----- + -- + ---- + ----- + ----- + ----- + ---- - 2 a z - 5 z - 6 4 2 9 7 5 3 a a a a a a a a 4 4 4 4 4 5 5 5 5 5 2 z 5 z 31 z 31 z 2 z 10 z 30 z 33 z 20 z 6 z > ---- + ---- + ----- + ----- + ---- - ----- - ----- - ----- - ----- - ---- + 10 8 6 4 2 9 7 5 3 a a a a a a a a a a 6 6 6 6 6 7 7 7 5 6 z 12 z 39 z 35 z 6 z 4 z 4 z 5 z > a z + 3 z + --- - ----- - ----- - ----- - ---- + ---- + ---- + ---- + 10 8 6 4 2 9 7 3 a a a a a a a a 7 8 8 8 8 9 9 9 10 10 5 z 6 z 15 z 15 z 6 z 4 z 8 z 4 z z z > ---- + ---- + ----- + ----- + ---- + ---- + ---- + ---- + --- + --- a 8 6 4 2 7 5 3 6 4 a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 18]], Vassiliev[3][Knot[11, Alternating, 18]]}
Out[15]=	{4, 5}
In[16]:=	Kh[Knot[11, Alternating, 18]][q, t]
Out[16]=	3 1 2 1 5 2 q 3 5 5 2 8 q + 6 q + ----- + ----- + ---- + --- + --- + 10 q t + 7 q t + 11 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 13 5 > 10 q t + 9 q t + 11 q t + 9 q t + 9 q t + 5 q t + 9 q t + 13 6 15 6 15 7 17 7 19 8 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a18

K11a17 K11a19