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

K11n27 K11n29

Knotscape

This page is passe. Go here instead!    The Knot K11n28 Visit K11n28's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X4251 X8493 X12,5,13,6 X2837 X9,15,10,14 X11,18,12,19 X6,13,7,14 X15,1,16,22 X17,21,18,20 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, 9, -11, 8}

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-2 - 5t-1 + 9 - 5t + t2

Conway Polynomial:

1 - z2 + z4

Other knots with the same Alexander/Conway Polynomial:

{77, ...}

Determinant and Signature:

{21, 0}

Jones Polynomial:

q-2 - q-1 + 2 - 3q + 3q2 - 3q3 + 3q4 - 2q5 + 2q6 - q7

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

{K11n64, ...}

A2 (sl(3)) Invariant:

q-8 + q-6 + q-2 - q2 - q4 - q6 + q10 + q12 + 2q14 - q22

HOMFLY-PT Polynomial:

- a-6 - a-6z2 + 3a-4 + 3a-4z2 + a-4z4 - 2a-2 - 2a-2z2 - z2 + a2

Kauffman Polynomial:

- a-7z + 6a-7z3 - 5a-7z5 + a-7z7 + a-6 - 8a-6z2 + 17a-6z4 - 11a-6z6 + 2a-6z8 + 2a-5z3 + 2a-5z5 - 4a-5z7 + a-5z9 + 3a-4 - 17a-4z2 + 26a-4z4 - 16a-4z6 + 3a-4z8 + 4a-3z - 10a-3z3 + 9a-3z5 - 5a-3z7 + a-3z9 + 2a-2 - 10a-2z2 + 10a-2z4 - 5a-2z6 + a-2z8 + 3a-1z - 5a-1z3 + 2a-1z5 + z4 + az3 - a2 + a2z2

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

{-1, 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 1128. 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 = 15                   1 j = 13                 1   j = 11               1 1   j = 9             2 1     j = 7           1 1       j = 5         2 2         j = 3       1 1           j = 1     1 2             j = -1   1 2               j = -3                     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, NonAlternating, 28]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 28]]
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, 1, 16, 22], > X[17, 21, 18, 20], X[19, 10, 20, 11], X[21, 17, 22, 16]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 28]]
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, NonAlternating, 28]]
Out[5]=	DTCode[4, 8, 12, 2, -14, -18, 6, -22, -20, -10, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 28]][t]
Out[6]=	-2 5 2 9 + t - - - 5 t + t t
In[7]:=	Conway[Knot[11, NonAlternating, 28]][z]
Out[7]=	2 4 1 - z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[7, 7], Knot[11, NonAlternating, 28]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 28]], KnotSignature[Knot[11, NonAlternating, 28]]}
Out[9]=	{21, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 28]][q]
Out[10]=	-2 1 2 3 4 5 6 7 2 + q - - - 3 q + 3 q - 3 q + 3 q - 2 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 28], Knot[11, NonAlternating, 64]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 28]][q]
Out[12]=	-8 -6 -2 2 4 6 10 12 14 22 q + q + q - q - q - q + q + q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 28]][a, z]
Out[13]=	2 2 2 4 -6 3 2 2 2 z 3 z 2 z z -a + -- - -- + a - z - -- + ---- - ---- + -- 4 2 6 4 2 4 a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 28]][a, z]
Out[14]=	2 2 2 3 -6 3 2 2 z 4 z 3 z 8 z 17 z 10 z 2 2 6 z a + -- + -- - a - -- + --- + --- - ---- - ----- - ----- + a z + ---- + 4 2 7 3 a 6 4 2 7 a a a a a a a a 3 3 3 4 4 4 5 5 2 z 10 z 5 z 3 4 17 z 26 z 10 z 5 z 2 z > ---- - ----- - ---- + a z + z + ----- + ----- + ----- - ---- + ---- + 5 3 a 6 4 2 7 5 a a a a a a a 5 5 6 6 6 7 7 7 8 8 8 9 z 2 z 11 z 16 z 5 z z 4 z 5 z 2 z 3 z z > ---- + ---- - ----- - ----- - ---- + -- - ---- - ---- + ---- + ---- + -- + 3 a 6 4 2 7 5 3 6 4 2 a a a a a a a a a a 9 9 z z > -- + -- 5 3 a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 28]], Vassiliev[3][Knot[11, NonAlternating, 28]]}
Out[15]=	{-1, 1}
In[16]:=	Kh[Knot[11, NonAlternating, 28]][q, t]
Out[16]=	2 1 1 3 3 2 5 2 5 3 7 3 - + q + ----- + --- + 2 q t + q t + q t + 2 q t + 2 q t + q t + q 5 2 q t q t 7 4 9 4 9 5 11 5 11 6 13 6 15 7 > q t + 2 q t + q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n28

K11n27 K11n29