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

K11n52 K11n54

Knotscape

This page is passe. Go here instead!    The Knot K11n53 Visit K11n53's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X4251 X8493 X5,14,6,15 X2837 X9,16,10,17 X11,19,12,18 X13,6,14,7 X15,22,16,1 X17,20,18,21 X19,11,20,10 X21,12,22,13

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 4t-2 - 8t-1 + 11 - 8t + 4t2 - t3

Conway Polynomial:

1 - z2 - 2z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{817, ...}

Determinant and Signature:

{37, 0}

Jones Polynomial:

q-6 - 2q-5 + 3q-4 - 5q-3 + 6q-2 - 6q-1 + 6 - 4q + 3q2 - q3

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

{914, ...}

A2 (sl(3)) Invariant:

q-18 + q-14 - 2q-10 - 2q-6 + q-2 + 1 + 3q2 + q6 - q10

HOMFLY-PT Polynomial:

- a-2 - a-2z2 + 5 + 6z2 + 2z4 - 5a2 - 9a2z2 - 5a2z4 - a2z6 + 2a4 + 3a4z2 + a4z4

Kauffman Polynomial:

- a-3z + a-3z3 + a-2 - 4a-2z2 + 3a-2z4 - 3a-1z + 5a-1z3 - 2a-1z5 + a-1z7 + 5 - 16z2 + 20z4 - 9z6 + 2z8 - 5az + 8az3 - az5 - 2az7 + az9 + 5a2 - 19a2z2 + 27a2z4 - 17a2z6 + 4a2z8 - 6a3z + 12a3z3 - 7a3z5 - a3z7 + a3z9 + 2a4 - 4a4z2 + 6a4z4 - 7a4z6 + 2a4z8 - 3a5z + 8a5z3 - 8a5z5 + 2a5z7 + 3a6z2 - 4a6z4 + a6z6

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

{-1, 2}

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 1153. 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 j = 7                   1 j = 5                 2   j = 3               2 1   j = 1             4 2     j = -1           3 3       j = -3         3 3         j = -5       2 3           j = -7     1 3             j = -9   1 2               j = -11   1                 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, NonAlternating, 53]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 53]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 14, 6, 15], X[2, 8, 3, 7], > X[9, 16, 10, 17], X[11, 19, 12, 18], X[13, 6, 14, 7], X[15, 22, 16, 1], > X[17, 20, 18, 21], X[19, 11, 20, 10], X[21, 12, 22, 13]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 53]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 7, 4, -2, -5, 10, -6, 11, -7, 3, -8, 5, -9, 6, -10, > 9, -11, 8]
In[5]:=	DTCode[Knot[11, NonAlternating, 53]]
Out[5]=	DTCode[4, 8, -14, 2, -16, -18, -6, -22, -20, -10, -12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 53]][t]
Out[6]=	-3 4 8 2 3 11 - t + -- - - - 8 t + 4 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 53]][z]
Out[7]=	2 4 6 1 - z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 17], Knot[11, NonAlternating, 53]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 53]], KnotSignature[Knot[11, NonAlternating, 53]]}
Out[9]=	{37, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 53]][q]
Out[10]=	-6 2 3 5 6 6 2 3 6 + q - -- + -- - -- + -- - - - 4 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[9, 14], Knot[11, NonAlternating, 53]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 53]][q]
Out[12]=	-18 -14 2 2 -2 2 6 10 1 + q + q - --- - -- + q + 3 q + q - q 10 6 q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 53]][a, z]
Out[13]=	2 -2 2 4 2 z 2 2 4 2 4 2 4 5 - a - 5 a + 2 a + 6 z - -- - 9 a z + 3 a z + 2 z - 5 a z + 2 a 4 4 2 6 > a z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 53]][a, z]
Out[14]=	2 -2 2 4 z 3 z 3 5 2 4 z 5 + a + 5 a + 2 a - -- - --- - 5 a z - 6 a z - 3 a z - 16 z - ---- - 3 a 2 a a 3 3 2 2 4 2 6 2 z 5 z 3 3 3 5 3 > 19 a z - 4 a z + 3 a z + -- + ---- + 8 a z + 12 a z + 8 a z + 3 a a 4 5 4 3 z 2 4 4 4 6 4 2 z 5 3 5 > 20 z + ---- + 27 a z + 6 a z - 4 a z - ---- - a z - 7 a z - 2 a a 7 5 5 6 2 6 4 6 6 6 z 7 3 7 > 8 a z - 9 z - 17 a z - 7 a z + a z + -- - 2 a z - a z + a 5 7 8 2 8 4 8 9 3 9 > 2 a z + 2 z + 4 a z + 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 53]], Vassiliev[3][Knot[11, NonAlternating, 53]]}
Out[15]=	{-1, 2}
In[16]:=	Kh[Knot[11, NonAlternating, 53]][q, t]
Out[16]=	3 1 1 1 2 1 3 2 3 - + 4 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 3 3 3 3 3 2 5 2 7 3 > ----- + ---- + --- + 2 q t + 2 q t + q t + 2 q t + q t 3 2 3 q t q t q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n53

K11n52 K11n54