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

K11n48 K11n50

Knotscape

This page is passe. Go here instead!    The Knot K11n49 Visit K11n49's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X4251 X8394 X5,12,6,13 X7,17,8,16 X2,9,3,10 X11,19,12,18 X13,22,14,1 X15,20,16,21 X17,11,18,10 X19,7,20,6 X21,14,22,15

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-2 + 3 - t2

Conway Polynomial:

1 - 4z2 - z4

Other knots with the same Alexander/Conway Polynomial:

{K11n116, ...}

Determinant and Signature:

{1, 0}

Jones Polynomial:

q-5 - q-4 + q-3 - q-2 + 1 - q + q2 - q3 + q4

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

{...}

A2 (sl(3)) Invariant:

q-16 + q-14 + q-12 - q-8 - q-6 - 2q-4 + q-2 + 1 + q2 - q6 + q12 + q14

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-4 - 3a-4z2 + a-4z4 + a-3z - 3a-3z3 + a-3z5 + a-2 - 3a-2z2 + a-2z4 + a-1z - a-1z3 + 2 - 7z2 + 13z4 - 7z6 + z8 + 3az - 10az3 + 14az5 - 7az7 + az9 + 3a2 - 18a2z2 + 28a2z4 - 14a2z6 + 2a2z8 + 3a3z - 12a3z3 + 15a3z5 - 7a3z7 + a3z9 + 2a4 - 11a4z2 + 15a4z4 - 7a4z6 + a4z8

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

{-4, 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 1149. 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 j = 9                     1 j = 7                       j = 5                 1 1   j = 3             1 1       j = 1             1 1       j = -1         1 2 2         j = -3       1               j = -5       1 1             j = -7   1 1                 j = -9                       j = -11 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, 49]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 49]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[5, 12, 6, 13], X[7, 17, 8, 16], > X[2, 9, 3, 10], X[11, 19, 12, 18], X[13, 22, 14, 1], X[15, 20, 16, 21], > X[17, 11, 18, 10], X[19, 7, 20, 6], X[21, 14, 22, 15]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 49]]
Out[4]=	GaussCode[1, -5, 2, -1, -3, 10, -4, -2, 5, 9, -6, 3, -7, 11, -8, 4, -9, 6, -10, > 8, -11, 7]
In[5]:=	DTCode[Knot[11, NonAlternating, 49]]
Out[5]=	DTCode[4, 8, -12, -16, 2, -18, -22, -20, -10, -6, -14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 49]][t]
Out[6]=	-2 2 3 - t - t
In[7]:=	Conway[Knot[11, NonAlternating, 49]][z]
Out[7]=	2 4 1 - 4 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 49], Knot[11, NonAlternating, 116]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 49]], KnotSignature[Knot[11, NonAlternating, 49]]}
Out[9]=	{1, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 49]][q]
Out[10]=	-5 -4 -3 -2 2 3 4 1 + q - q + q - q - q + q - q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 49]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 49]][q]
Out[12]=	-16 -14 -12 -8 -6 2 -2 2 6 12 14 1 + q + q + q - q - q - -- + q + q - q + q + q 4 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 49]][a, z]
Out[13]=	2 -4 -2 2 4 z 2 2 4 2 2 4 2 + a - a - 3 a + 2 a - -- - 4 a z + a z - a z 2 a
In[14]:=	Kauffman[Knot[11, NonAlternating, 49]][a, z]
Out[14]=	2 2 -4 -2 2 4 z z 3 2 3 z 3 z 2 + a + a + 3 a + 2 a + -- + - + 3 a z + 3 a z - 7 z - ---- - ---- - 3 a 4 2 a a a 3 3 4 4 2 2 4 2 3 z z 3 3 3 4 z z > 18 a z - 11 a z - ---- - -- - 10 a z - 12 a z + 13 z + -- + -- + 3 a 4 2 a a a 5 2 4 4 4 z 5 3 5 6 2 6 4 6 > 28 a z + 15 a z + -- + 14 a z + 15 a z - 7 z - 14 a z - 7 a z - 3 a 7 3 7 8 2 8 4 8 9 3 9 > 7 a z - 7 a z + z + 2 a z + a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 49]], Vassiliev[3][Knot[11, NonAlternating, 49]]}
Out[15]=	{-4, 1}
In[16]:=	Kh[Knot[11, NonAlternating, 49]][q, t]
Out[16]=	2 3 1 1 1 1 1 1 1 2 - + q + q + ------ + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 6 7 5 7 4 5 3 3 3 5 2 2 q t q t q t q t q t q t q t q t 3 5 2 5 3 9 4 > q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n49

K11n48 K11n50