K11a3

K11a2

K11a4

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a3 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X18,11,19,12 X6,14,7,13 X20,15,21,16 X22,17,1,18 X12,19,13,20 X16,21,17,22
Gauss code	1, -5, 2, -1, 3, -7, 4, -2, 5, -3, 6, -10, 7, -4, 8, -11, 9, -6, 10, -8, 11, -9
Dowker-Thistlethwaite code	4 8 10 14 2 18 6 20 22 12 16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a3/ThurstonBennequinNumber Hyperbolic Volume 15.55 A-Polynomial See Data:K11a3/A-polynomial

[edit Notes for K11a3's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant 2

[edit Notes for K11a3's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-13t^{2}+24t-29+24t^{-1}-13t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}-3z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 115, -2 }
Jones polynomial	$q^{3}-3q^{2}+7q-12+16q^{-1}-18q^{-2}+19q^{-3}-16q^{-4}+12q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+2a^{4}z^{6}-6a^{2}z^{6}+z^{6}-a^{6}z^{4}+9a^{4}z^{4}-15a^{2}z^{4}+4z^{4}-3a^{6}z^{2}+15a^{4}z^{2}-17a^{2}z^{2}+6z^{2}-3a^{6}+8a^{4}-7a^{2}+3$
Kauffman polynomial (db, data sources)	$a^{4}z^{10}+a^{2}z^{10}+4a^{5}z^{9}+8a^{3}z^{9}+4az^{9}+6a^{6}z^{8}+13a^{4}z^{8}+12a^{2}z^{8}+5z^{8}+5a^{7}z^{7}+a^{5}z^{7}-11a^{3}z^{7}-4az^{7}+3z^{7}a^{-1}+3a^{8}z^{6}-9a^{6}z^{6}-38a^{4}z^{6}-40a^{2}z^{6}+z^{6}a^{-2}-13z^{6}+a^{9}z^{5}-7a^{7}z^{5}-11a^{5}z^{5}-2a^{3}z^{5}-7az^{5}-8z^{5}a^{-1}-5a^{8}z^{4}+9a^{6}z^{4}+48a^{4}z^{4}+49a^{2}z^{4}-3z^{4}a^{-2}+12z^{4}-2a^{9}z^{3}+3a^{7}z^{3}+12a^{5}z^{3}+10a^{3}z^{3}+9az^{3}+6z^{3}a^{-1}+2a^{8}z^{2}-8a^{6}z^{2}-30a^{4}z^{2}-30a^{2}z^{2}+2z^{2}a^{-2}-8z^{2}+a^{9}z-a^{7}z-4a^{5}z-4a^{3}z-4az-2za^{-1}+3a^{6}+8a^{4}+7a^{2}+3$
The A2 invariant	$-q^{24}-q^{20}-2q^{18}+4q^{16}-q^{14}+3q^{12}+2q^{10}-2q^{8}+3q^{6}-5q^{4}+2q^{2}-1-q^{-2}+3q^{-4}-q^{-6}+q^{-8}$
The G2 invariant	$q^{128}-2q^{126}+5q^{124}-8q^{122}+9q^{120}-8q^{118}+q^{116}+13q^{114}-29q^{112}+47q^{110}-59q^{108}+53q^{106}-30q^{104}-20q^{102}+87q^{100}-150q^{98}+191q^{96}-186q^{94}+112q^{92}+17q^{90}-181q^{88}+324q^{86}-389q^{84}+334q^{82}-167q^{80}-81q^{78}+315q^{76}-446q^{74}+423q^{72}-237q^{70}-30q^{68}+264q^{66}-367q^{64}+285q^{62}-53q^{60}-216q^{58}+409q^{56}-407q^{54}+208q^{52}+130q^{50}-455q^{48}+647q^{46}-610q^{44}+353q^{42}+36q^{40}-415q^{38}+658q^{36}-671q^{34}+468q^{32}-122q^{30}-234q^{28}+454q^{26}-482q^{24}+308q^{22}-27q^{20}-243q^{18}+372q^{16}-315q^{14}+91q^{12}+196q^{10}-420q^{8}+474q^{6}-340q^{4}+65q^{2}+227-431q^{-2}+485q^{-4}-368q^{-6}+159q^{-8}+69q^{-10}-235q^{-12}+294q^{-14}-251q^{-16}+153q^{-18}-39q^{-20}-45q^{-22}+88q^{-24}-92q^{-26}+69q^{-28}-36q^{-30}+11q^{-32}+6q^{-34}-13q^{-36}+11q^{-38}-9q^{-40}+5q^{-42}-2q^{-44}+q^{-46}$

Further Quantum Invariants

K11a3 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a3"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-t^{4}+5t^{3}-13t^{2}+24t-29+24t^{-1}-13t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-3z^{6}-3z^{4}+z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 115, -2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{3}-3q^{2}+7q-12+16q^{-1}-18q^{-2}+19q^{-3}-16q^{-4}+12q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-a^{2}z^{8}+2a^{4}z^{6}-6a^{2}z^{6}+z^{6}-a^{6}z^{4}+9a^{4}z^{4}-15a^{2}z^{4}+4z^{4}-3a^{6}z^{2}+15a^{4}z^{2}-17a^{2}z^{2}+6z^{2}-3a^{6}+8a^{4}-7a^{2}+3$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$a^{4}z^{10}+a^{2}z^{10}+4a^{5}z^{9}+8a^{3}z^{9}+4az^{9}+6a^{6}z^{8}+13a^{4}z^{8}+12a^{2}z^{8}+5z^{8}+5a^{7}z^{7}+a^{5}z^{7}-11a^{3}z^{7}-4az^{7}+3z^{7}a^{-1}+3a^{8}z^{6}-9a^{6}z^{6}-38a^{4}z^{6}-40a^{2}z^{6}+z^{6}a^{-2}-13z^{6}+a^{9}z^{5}-7a^{7}z^{5}-11a^{5}z^{5}-2a^{3}z^{5}-7az^{5}-8z^{5}a^{-1}-5a^{8}z^{4}+9a^{6}z^{4}+48a^{4}z^{4}+49a^{2}z^{4}-3z^{4}a^{-2}+12z^{4}-2a^{9}z^{3}+3a^{7}z^{3}+12a^{5}z^{3}+10a^{3}z^{3}+9az^{3}+6z^{3}a^{-1}+2a^{8}z^{2}-8a^{6}z^{2}-30a^{4}z^{2}-30a^{2}z^{2}+2z^{2}a^{-2}-8z^{2}+a^{9}z-a^{7}z-4a^{5}z-4a^{3}z-4az-2za^{-1}+3a^{6}+8a^{4}+7a^{2}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a51, K11a331,}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a3"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ $-t^{4}+5t^{3}-13t^{2}+24t-29+24t^{-1}-13t^{-2}+5t^{-3}-t^{-4}$, $q^{3}-3q^{2}+7q-12+16q^{-1}-18q^{-2}+19q^{-3}-16q^{-4}+12q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11a51, K11a331,}

Vassiliev invariants

V2 and V3:

(1, -4)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $4$ $-32$ $8$ ${\frac {350}{3}}$ ${\frac {82}{3}}$ $-128$ $-{\frac {992}{3}}$ $-{\frac {128}{3}}$ $-32$ ${\frac {32}{3}}$ $512$ ${\frac {1400}{3}}$ ${\frac {328}{3}}$ ${\frac {49231}{30}}$ ${\frac {1618}{15}}$ ${\frac {21422}{45}}$ ${\frac {881}{18}}$ ${\frac {1231}{30}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). 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 K11a3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     2   -2 3                   5 1   4 1                 7 2     -5 -1               9 5       4 -3             10 8         -2 -5           9 8           1 -7         7 10             3 -9       5 9               -4 -11     2 7                 5 -13   1 5                   -4 -15   2                     2 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages. See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top.

K11a2 K11a4