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

K11n2 K11n4

Knotscape

This page is passe. Go here instead!    The Knot K11n3 Visit K11n3's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X7,15,8,14 X2,9,3,10 X11,16,12,17 X13,20,14,21 X15,7,16,6 X17,22,18,1 X19,12,20,13 X21,18,22,19

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- 3t-2 + 11t-1 - 15 + 11t - 3t2

Conway Polynomial:

1 - z2 - 3z4

Other knots with the same Alexander/Conway Polynomial:

{107, K11a59, ...}

Determinant and Signature:

{43, -2}

Jones Polynomial:

- q-6 + 3q-5 - 5q-4 + 7q-3 - 7q-2 + 7q-1 - 6 + 4q - 2q2 + q3

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

{922, ...}

A2 (sl(3)) Invariant:

- q-20 - q-18 + 2q-16 + q-12 + 2q-10 - q-8 + q-6 - 2q-4 - q2 + 2q4 + q10

HOMFLY-PT Polynomial:

a-2 + a-2z2 - z2 - z4 - 2a2 - 4a2z2 - 2a2z4 + 3a4 + 3a4z2 - a6

Kauffman Polynomial:

- a-2 + 4a-2z2 - 4a-2z4 + a-2z6 - 2a-1z + 6a-1z3 - 7a-1z5 + 2a-1z7 + 2z2 - 5z6 + 2z8 - 3az + 7az3 - 7az5 + az9 + 2a2 - 10a2z2 + 16a2z4 - 13a2z6 + 4a2z8 - 2a3z + 4a3z3 - a3z5 - a3z7 + a3z9 + 3a4 - 11a4z2 + 15a4z4 - 7a4z6 + 2a4z8 - 2a5z + 4a5z3 - a5z5 + a5z7 + a6 - 3a6z2 + 3a6z4 - a7z + a7z3

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=-2 is the signature of 113. 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 j = 7                   1 j = 5                 1   j = 3               3 1   j = 1             3 1     j = -1           4 3       j = -3         4 4         j = -5       3 3           j = -7     2 4             j = -9   1 3               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, NonAlternating, 3]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 3]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[7, 15, 8, 14], > X[2, 9, 3, 10], X[11, 16, 12, 17], X[13, 20, 14, 21], X[15, 7, 16, 6], > X[17, 22, 18, 1], X[19, 12, 20, 13], X[21, 18, 22, 19]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 3]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, 8, -4, -2, 5, -3, -6, 10, -7, 4, -8, 6, -9, 11, -10, > 7, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 3]]
Out[5]=	DTCode[4, 8, 10, -14, 2, -16, -20, -6, -22, -12, -18]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 3]][t]
Out[6]=	3 11 2 -15 - -- + -- + 11 t - 3 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 3]][z]
Out[7]=	2 4 1 - z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 7], Knot[11, Alternating, 59], Knot[11, NonAlternating, 3]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 3]], KnotSignature[Knot[11, NonAlternating, 3]]}
Out[9]=	{43, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 3]][q]
Out[10]=	-6 3 5 7 7 7 2 3 -6 - q + -- - -- + -- - -- + - + 4 q - 2 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, 22], Knot[11, NonAlternating, 3]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 3]][q]
Out[12]=	-20 -18 2 -12 2 -8 -6 2 2 4 10 -q - q + --- + q + --- - q + q - -- - q + 2 q + q 16 10 4 q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 3]][a, z]
Out[13]=	2 -2 2 4 6 2 z 2 2 4 2 4 2 4 a - 2 a + 3 a - a - z + -- - 4 a z + 3 a z - z - 2 a z 2 a
In[14]:=	Kauffman[Knot[11, NonAlternating, 3]][a, z]
Out[14]=	2 -2 2 4 6 2 z 3 5 7 2 4 z -a + 2 a + 3 a + a - --- - 3 a z - 2 a z - 2 a z - a z + 2 z + ---- - a 2 a 3 2 2 4 2 6 2 6 z 3 3 3 5 3 7 3 > 10 a z - 11 a z - 3 a z + ---- + 7 a z + 4 a z + 4 a z + a z - a 4 5 4 z 2 4 4 4 6 4 7 z 5 3 5 5 5 > ---- + 16 a z + 15 a z + 3 a z - ---- - 7 a z - a z - a z - 2 a a 6 7 6 z 2 6 4 6 2 z 3 7 5 7 8 2 8 > 5 z + -- - 13 a z - 7 a z + ---- - a z + a z + 2 z + 4 a z + 2 a a 4 8 9 3 9 > 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 3]], Vassiliev[3][Knot[11, NonAlternating, 3]]}
Out[15]=	{-1, -1}
In[16]:=	Kh[Knot[11, NonAlternating, 3]][q, t]
Out[16]=	4 4 1 2 1 3 2 4 3 3 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q q t q t q t q t q t q t q t q t 4 3 t 2 3 2 3 3 5 3 7 4 > ---- + --- + 3 q t + q t + 3 q t + q t + q t + q t 3 q q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n3

K11n2 K11n4