9 47

9_46

9_48

Visit 9 47's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 47's page at Knotilus! Visit 9 47's page at the original Knot Atlas! 9 47 Quick Notes

Simple square depiction

Threefold symmetrical depiction

Ornate threefold symmetrical depiction

Knot presentations

Planar diagram presentation	X6271 X16,8,17,7 X8394 X2,15,3,16 X14,9,15,10 X10,6,11,5 X4,14,5,13 X11,1,12,18 X17,13,18,12
Gauss code	1, -4, 3, -7, 6, -1, 2, -3, 5, -6, -8, 9, 7, -5, 4, -2, -9, 8
Dowker-Thistlethwaite code	6 8 10 16 14 -18 4 2 -12
Conway Notation	[8*-20]

Minimum Braid Representative: Length is 9, width is 4. Braid index is 4.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index $\{4,6\}$ Nakanishi index 2 Maximal Thurston-Bennequin number [-2][-7] Hyperbolic Volume 10.05 A-Polynomial See Data:9 47/A-polynomial

[edit Notes for 9 47's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant -2

[edit Notes for 9 47's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-4t^{2}+6t-5+6t^{-1}-4t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+2z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{3,t+1\}$
Determinant and Signature	{ 27, 2 }
Jones polynomial	$2q^{5}-4q^{4}+4q^{3}-5q^{2}+5q-3+3q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+4z^{4}a^{-2}-z^{4}a^{-4}-z^{4}+4z^{2}a^{-2}-3z^{2}a^{-4}-2z^{2}+a^{-2}-2a^{-4}+a^{-6}+1$
Kauffman polynomial (db, data sources)	$2z^{7}a^{-1}+2z^{7}a^{-3}+6z^{6}a^{-2}+3z^{6}a^{-4}+3z^{6}+az^{5}-4z^{5}a^{-1}-4z^{5}a^{-3}+z^{5}a^{-5}-16z^{4}a^{-2}-7z^{4}a^{-4}-9z^{4}-2az^{3}+z^{3}a^{-1}+6z^{3}a^{-3}+3z^{3}a^{-5}+11z^{2}a^{-2}+9z^{2}a^{-4}+3z^{2}a^{-6}+5z^{2}-2za^{-1}-5za^{-3}-3za^{-5}-a^{-2}-2a^{-4}-a^{-6}+1$
The A2 invariant	$-q^{6}+q^{4}+q^{2}+2+2q^{-2}-q^{-4}+q^{-6}-2q^{-8}-q^{-12}-q^{-14}+q^{-16}+q^{-20}$
The G2 invariant	$q^{32}-2q^{30}+4q^{28}-7q^{26}+4q^{24}-q^{22}-8q^{20}+16q^{18}-16q^{16}+16q^{14}-4q^{12}-11q^{10}+20q^{8}-18q^{6}+14q^{4}-2q^{2}-13+21q^{-2}-8q^{-4}+13q^{-8}-19q^{-10}+18q^{-12}-4q^{-14}-9q^{-16}+12q^{-18}-21q^{-20}+25q^{-22}-13q^{-24}+9q^{-28}-19q^{-30}+23q^{-32}-18q^{-34}+4q^{-36}+3q^{-38}-14q^{-40}+19q^{-42}-11q^{-44}-3q^{-46}+15q^{-48}-19q^{-50}+12q^{-52}+q^{-54}-17q^{-56}+20q^{-58}-17q^{-60}+11q^{-62}+2q^{-64}-11q^{-66}+14q^{-68}-12q^{-70}+8q^{-72}-5q^{-76}+2q^{-78}-2q^{-80}+2q^{-82}-q^{-84}+2q^{-86}+q^{-88}$

Further Quantum Invariants

Further quantum knot invariants for 9_47.

A1 Invariants.

Weight	Invariant
1	$-q^{5}+2q^{3}+2q^{-1}-q^{-5}-2q^{-9}+2q^{-11}$
2	$q^{16}-2q^{14}-3q^{12}+5q^{10}+2q^{8}-5q^{6}+3q^{4}+6q^{2}-4-q^{-2}+4q^{-4}-3q^{-6}-4q^{-8}+2q^{-10}+2q^{-12}-3q^{-14}-q^{-16}+7q^{-18}-q^{-20}-5q^{-22}+6q^{-24}-5q^{-28}+q^{-30}+q^{-32}$
3	$-q^{33}+2q^{31}+3q^{29}-2q^{27}-8q^{25}-5q^{23}+11q^{21}+11q^{19}-5q^{17}-17q^{15}-2q^{13}+20q^{11}+15q^{9}-14q^{7}-20q^{5}+6q^{3}+24q-q^{-1}-24q^{-3}-7q^{-5}+19q^{-7}+10q^{-9}-18q^{-11}-8q^{-13}+13q^{-15}+12q^{-17}-9q^{-19}-10q^{-21}+4q^{-23}+16q^{-25}+3q^{-27}-16q^{-29}-13q^{-31}+13q^{-33}+21q^{-35}-10q^{-37}-27q^{-39}+q^{-41}+27q^{-43}+5q^{-45}-17q^{-47}-12q^{-49}+11q^{-51}+10q^{-53}-3q^{-55}-4q^{-57}-2q^{-59}+2q^{-61}$
4	$q^{56}-2q^{54}-3q^{52}+2q^{50}+5q^{48}+11q^{46}-2q^{44}-16q^{42}-18q^{40}-10q^{38}+29q^{36}+33q^{34}+10q^{32}-26q^{30}-58q^{28}-16q^{26}+35q^{24}+70q^{22}+43q^{20}-51q^{18}-83q^{16}-45q^{14}+56q^{12}+109q^{10}+32q^{8}-71q^{6}-116q^{4}-24q^{2}+98+97q^{-2}-5q^{-4}-109q^{-6}-72q^{-8}+49q^{-10}+97q^{-12}+32q^{-14}-69q^{-16}-68q^{-18}+17q^{-20}+70q^{-22}+30q^{-24}-42q^{-26}-53q^{-28}+54q^{-32}+32q^{-34}-20q^{-36}-52q^{-38}-34q^{-40}+32q^{-42}+60q^{-44}+38q^{-46}-47q^{-48}-95q^{-50}-23q^{-52}+75q^{-54}+116q^{-56}+9q^{-58}-120q^{-60}-102q^{-62}+18q^{-64}+135q^{-66}+84q^{-68}-58q^{-70}-108q^{-72}-55q^{-74}+60q^{-76}+84q^{-78}+16q^{-80}-38q^{-82}-49q^{-84}-6q^{-86}+22q^{-88}+17q^{-90}+4q^{-92}-8q^{-94}-5q^{-96}+q^{-98}+q^{-100}$
5	$-q^{85}+2q^{83}+3q^{81}-2q^{79}-5q^{77}-8q^{75}-4q^{73}+7q^{71}+24q^{69}+24q^{67}+3q^{65}-27q^{63}-51q^{61}-48q^{59}-5q^{57}+66q^{55}+95q^{53}+67q^{51}-13q^{49}-112q^{47}-156q^{45}-90q^{43}+58q^{41}+190q^{39}+220q^{37}+93q^{35}-137q^{33}-303q^{31}-264q^{29}-17q^{27}+278q^{25}+409q^{23}+230q^{21}-146q^{19}-452q^{17}-434q^{15}-65q^{13}+380q^{11}+548q^{9}+284q^{7}-221q^{5}-572q^{3}-451q+40q^{-1}+495q^{-3}+539q^{-5}+130q^{-7}-371q^{-9}-539q^{-11}-236q^{-13}+254q^{-15}+480q^{-17}+282q^{-19}-146q^{-21}-402q^{-23}-276q^{-25}+86q^{-27}+317q^{-29}+229q^{-31}-48q^{-33}-254q^{-35}-192q^{-37}+48q^{-39}+206q^{-41}+151q^{-43}-34q^{-45}-183q^{-47}-159q^{-49}+13q^{-51}+173q^{-53}+191q^{-55}+55q^{-57}-150q^{-59}-255q^{-61}-160q^{-63}+92q^{-65}+319q^{-67}+312q^{-69}+19q^{-71}-345q^{-73}-469q^{-75}-191q^{-77}+296q^{-79}+590q^{-81}+398q^{-83}-168q^{-85}-624q^{-87}-572q^{-89}-30q^{-91}+534q^{-93}+663q^{-95}+242q^{-97}-364q^{-99}-619q^{-101}-370q^{-103}+129q^{-105}+472q^{-107}+404q^{-109}+42q^{-111}-276q^{-113}-319q^{-115}-135q^{-117}+99q^{-119}+199q^{-121}+136q^{-123}+q^{-125}-89q^{-127}-78q^{-129}-31q^{-131}+18q^{-133}+35q^{-135}+18q^{-137}-3q^{-139}-6q^{-141}-4q^{-143}-2q^{-145}+2q^{-147}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{6}+q^{4}+q^{2}+2+2q^{-2}-q^{-4}+q^{-6}-2q^{-8}-q^{-12}-q^{-14}+q^{-16}+q^{-20}$
1,1	$q^{20}-4q^{18}+10q^{16}-22q^{14}+32q^{12}-46q^{10}+56q^{8}-52q^{6}+48q^{4}-22q^{2}+4+36q^{-2}-58q^{-4}+74q^{-6}-94q^{-8}+90q^{-10}-94q^{-12}+70q^{-14}-52q^{-16}+28q^{-18}+5q^{-20}-24q^{-22}+50q^{-24}-56q^{-26}+58q^{-28}-46q^{-30}+32q^{-32}-22q^{-34}+10q^{-36}-2q^{-38}-4q^{-40}+2q^{-46}$
2,0	$q^{18}-q^{16}-3q^{14}-q^{12}+2q^{10}+2q^{8}+4q^{4}+6q^{2}+3-2q^{-2}+q^{-4}-3q^{-6}-3q^{-8}-2q^{-10}-2q^{-12}-2q^{-14}-2q^{-16}+3q^{-18}+3q^{-24}+5q^{-26}-q^{-28}-q^{-30}+q^{-32}-2q^{-38}-q^{-48}+q^{-52}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{14}-2q^{12}+q^{8}-4q^{6}+5q^{4}+3q^{2}+1+7q^{-2}+3q^{-4}-4q^{-6}-2q^{-8}-3q^{-10}-5q^{-12}-2q^{-14}+5q^{-18}+q^{-20}+q^{-22}+5q^{-24}-4q^{-26}-3q^{-28}+3q^{-30}-3q^{-32}-q^{-34}+3q^{-36}$
1,0,0	$-q^{7}+q^{5}+3q+q^{-1}+2q^{-3}-q^{-11}-2q^{-15}-2q^{-19}+q^{-21}+q^{-25}+q^{-27}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{16}-q^{14}-q^{12}-2q^{8}+4q^{4}+3q^{2}+3+9q^{-2}+7q^{-4}+q^{-6}-2q^{-8}+q^{-10}-6q^{-12}-12q^{-14}-5q^{-16}-6q^{-20}+q^{-22}+9q^{-24}+2q^{-26}+2q^{-28}+6q^{-30}+3q^{-32}-4q^{-34}-q^{-36}+2q^{-38}-3q^{-40}-5q^{-42}+q^{-44}+2q^{-46}-2q^{-48}+q^{-50}+2q^{-52}$
1,0,0,0	$-q^{8}+q^{6}+2q^{2}+2+q^{-2}+2q^{-4}+q^{-8}-q^{-10}+q^{-12}-q^{-14}-2q^{-18}-q^{-20}-q^{-22}-2q^{-24}+q^{-26}+q^{-30}+q^{-32}+q^{-34}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{14}+2q^{12}-4q^{10}+5q^{8}-4q^{6}+5q^{4}-3q^{2}+3+q^{-2}-q^{-4}+6q^{-6}-6q^{-8}+9q^{-10}-9q^{-12}+8q^{-14}-8q^{-16}+3q^{-18}-3q^{-20}-q^{-22}+q^{-24}-4q^{-26}+5q^{-28}-5q^{-30}+5q^{-32}-3q^{-34}+3q^{-36}$
1,0	$q^{24}-2q^{20}-2q^{18}+2q^{16}+3q^{14}-3q^{12}-4q^{10}+q^{8}+7q^{6}+3q^{4}-3q^{2}-1+6q^{-2}+5q^{-4}+q^{-6}-4q^{-8}-q^{-10}+q^{-12}-q^{-14}-6q^{-16}-4q^{-18}+q^{-20}+q^{-22}-2q^{-24}-4q^{-26}+3q^{-28}+6q^{-30}+2q^{-32}-3q^{-34}+q^{-36}+5q^{-38}+3q^{-40}-4q^{-42}-5q^{-44}+q^{-46}+4q^{-48}-q^{-50}-4q^{-52}-2q^{-54}+q^{-56}+3q^{-58}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{18}-2q^{16}+2q^{14}-4q^{12}+4q^{10}-4q^{8}+4q^{6}-2q^{4}+6q^{2}+2+3q^{-2}+4q^{-4}+3q^{-8}-6q^{-10}+4q^{-12}-9q^{-14}+4q^{-16}-9q^{-18}+5q^{-20}-5q^{-22}+6q^{-24}-2q^{-26}+2q^{-28}+q^{-30}+2q^{-34}-4q^{-36}+2q^{-38}-5q^{-40}+5q^{-42}-3q^{-44}+2q^{-46}-2q^{-48}+3q^{-50}$

G2 Invariants.

Weight

Invariant

1,0

$q^{32}-2q^{30}+4q^{28}-7q^{26}+4q^{24}-q^{22}-8q^{20}+16q^{18}-16q^{16}+16q^{14}-4q^{12}-11q^{10}+20q^{8}-18q^{6}+14q^{4}-2q^{2}-13+21q^{-2}-8q^{-4}+13q^{-8}-19q^{-10}+18q^{-12}-4q^{-14}-9q^{-16}+12q^{-18}-21q^{-20}+25q^{-22}-13q^{-24}+9q^{-28}-19q^{-30}+23q^{-32}-18q^{-34}+4q^{-36}+3q^{-38}-14q^{-40}+19q^{-42}-11q^{-44}-3q^{-46}+15q^{-48}-19q^{-50}+12q^{-52}+q^{-54}-17q^{-56}+20q^{-58}-17q^{-60}+11q^{-62}+2q^{-64}-11q^{-66}+14q^{-68}-12q^{-70}+8q^{-72}-5q^{-76}+2q^{-78}-2q^{-80}+2q^{-82}-q^{-84}+2q^{-86}+q^{-88}$

.

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["9 47"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-4t^{2}+6t-5+6t^{-1}-4t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\{3,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 27, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$2q^{5}-4q^{4}+4q^{3}-5q^{2}+5q-3+3q^{-1}-q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{7}a^{-1}+2z^{7}a^{-3}+6z^{6}a^{-2}+3z^{6}a^{-4}+3z^{6}+az^{5}-4z^{5}a^{-1}-4z^{5}a^{-3}+z^{5}a^{-5}-16z^{4}a^{-2}-7z^{4}a^{-4}-9z^{4}-2az^{3}+z^{3}a^{-1}+6z^{3}a^{-3}+3z^{3}a^{-5}+11z^{2}a^{-2}+9z^{2}a^{-4}+3z^{2}a^{-6}+5z^{2}-2za^{-1}-5za^{-3}-3za^{-5}-a^{-2}-2a^{-4}-a^{-6}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

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

Vassiliev invariants

V2 and V3:

(-1, -2)

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 $-4$ $-16$ $8$ $-{\frac {110}{3}}$ $-{\frac {34}{3}}$ $64$ ${\frac {32}{3}}$ ${\frac {32}{3}}$ $-16$ $-{\frac {32}{3}}$ $128$ ${\frac {440}{3}}$ ${\frac {136}{3}}$ ${\frac {10529}{30}}$ $-{\frac {418}{15}}$ ${\frac {10738}{45}}$ $-{\frac {833}{18}}$ ${\frac {1409}{30}}$

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 9 47. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 χ 11               2 2 9             2   -2 7           2 2   0 5         3 2     -1 3       2 2       0 1     2 4         2 -1   1 1           0 -3   2             2 -5 1               -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{15}-6q^{13}+6q^{12}+6q^{11}-17q^{10}+10q^{9}+14q^{8}-25q^{7}+8q^{6}+19q^{5}-25q^{4}+2q^{3}+20q^{2}-18q-3+17q^{-1}-8q^{-2}-6q^{-3}+9q^{-4}-q^{-5}-3q^{-6}+q^{-7}$
3	$2q^{29}-4q^{28}-2q^{27}+q^{26}+15q^{25}-3q^{24}-25q^{23}-4q^{22}+37q^{21}+19q^{20}-51q^{19}-32q^{18}+54q^{17}+50q^{16}-59q^{15}-58q^{14}+51q^{13}+69q^{12}-46q^{11}-70q^{10}+37q^{9}+70q^{8}-25q^{7}-69q^{6}+16q^{5}+60q^{4}+3q^{3}-60q^{2}-10q+43+26q^{-1}-35q^{-2}-28q^{-3}+17q^{-4}+32q^{-5}-6q^{-6}-23q^{-7}-5q^{-8}+17q^{-9}+6q^{-10}-7q^{-11}-5q^{-12}+q^{-13}+3q^{-14}-q^{-15}$
4	$q^{48}-6q^{46}-3q^{45}+12q^{44}+14q^{43}+5q^{42}-34q^{41}-46q^{40}+23q^{39}+68q^{38}+73q^{37}-58q^{36}-161q^{35}-30q^{34}+118q^{33}+215q^{32}-7q^{31}-278q^{30}-150q^{29}+100q^{28}+344q^{27}+100q^{26}-319q^{25}-248q^{24}+28q^{23}+392q^{22}+185q^{21}-297q^{20}-276q^{19}-38q^{18}+374q^{17}+217q^{16}-245q^{15}-254q^{14}-92q^{13}+321q^{12}+228q^{11}-173q^{10}-214q^{9}-145q^{8}+236q^{7}+227q^{6}-72q^{5}-149q^{4}-193q^{3}+115q^{2}+190q+32-47q^{-1}-192q^{-2}-7q^{-3}+98q^{-4}+77q^{-5}+56q^{-6}-115q^{-7}-60q^{-8}-3q^{-9}+39q^{-10}+88q^{-11}-21q^{-12}-33q^{-13}-38q^{-14}-12q^{-15}+46q^{-16}+11q^{-17}+3q^{-18}-15q^{-19}-16q^{-20}+7q^{-21}+3q^{-22}+5q^{-23}-q^{-24}-3q^{-25}+q^{-26}$
5	$2q^{71}-4q^{70}-2q^{69}-2q^{68}+3q^{67}+21q^{66}+19q^{65}-21q^{64}-51q^{63}-49q^{62}-8q^{61}+111q^{60}+154q^{59}+42q^{58}-151q^{57}-283q^{56}-192q^{55}+154q^{54}+472q^{53}+404q^{52}-83q^{51}-626q^{50}-691q^{49}-95q^{48}+727q^{47}+1010q^{46}+338q^{45}-755q^{44}-1255q^{43}-637q^{42}+675q^{41}+1466q^{40}+904q^{39}-563q^{38}-1549q^{37}-1124q^{36}+397q^{35}+1590q^{34}+1268q^{33}-270q^{32}-1542q^{31}-1351q^{30}+145q^{29}+1495q^{28}+1373q^{27}-65q^{26}-1406q^{25}-1369q^{24}-15q^{23}+1323q^{22}+1349q^{21}+84q^{20}-1221q^{19}-1314q^{18}-173q^{17}+1083q^{16}+1287q^{15}+290q^{14}-944q^{13}-1226q^{12}-404q^{11}+721q^{10}+1161q^{9}+546q^{8}-516q^{7}-1028q^{6}-630q^{5}+231q^{4}+858q^{3}+714q^{2}-15q-619-674q^{-1}-224q^{-2}+367q^{-3}+593q^{-4}+336q^{-5}-114q^{-6}-410q^{-7}-392q^{-8}-78q^{-9}+224q^{-10}+318q^{-11}+192q^{-12}-34q^{-13}-213q^{-14}-209q^{-15}-71q^{-16}+71q^{-17}+153q^{-18}+132q^{-19}+17q^{-20}-82q^{-21}-101q^{-22}-61q^{-23}+5q^{-24}+66q^{-25}+61q^{-26}+17q^{-27}-21q^{-28}-33q^{-29}-24q^{-30}-5q^{-31}+18q^{-32}+14q^{-33}+3q^{-34}-3q^{-35}-3q^{-36}-5q^{-37}+q^{-38}+3q^{-39}-q^{-40}$
6	$q^{99}-6q^{97}-3q^{96}+6q^{95}+11q^{94}+14q^{93}+12q^{92}-7q^{91}-64q^{90}-83q^{89}-18q^{88}+78q^{87}+162q^{86}+204q^{85}+92q^{84}-220q^{83}-477q^{82}-434q^{81}-65q^{80}+445q^{79}+935q^{78}+897q^{77}+32q^{76}-1075q^{75}-1656q^{74}-1225q^{73}+77q^{72}+1873q^{71}+2748q^{70}+1631q^{69}-829q^{68}-3059q^{67}-3524q^{66}-1819q^{65}+1792q^{64}+4634q^{63}+4325q^{62}+970q^{61}-3303q^{60}-5637q^{59}-4594q^{58}+244q^{57}+5213q^{56}+6574q^{55}+3386q^{54}-2186q^{53}-6367q^{52}-6680q^{51}-1701q^{50}+4511q^{49}+7418q^{48}+5032q^{47}-767q^{46}-5949q^{45}-7440q^{44}-2940q^{43}+3523q^{42}+7244q^{41}+5580q^{40}+139q^{39}-5253q^{38}-7358q^{37}-3392q^{36}+2818q^{35}+6777q^{34}+5549q^{33}+612q^{32}-4638q^{31}-7027q^{30}-3574q^{29}+2217q^{28}+6240q^{27}+5439q^{26}+1127q^{25}-3892q^{24}-6593q^{23}-3894q^{22}+1292q^{21}+5430q^{20}+5347q^{19}+1991q^{18}-2664q^{17}-5850q^{16}-4346q^{15}-152q^{14}+4030q^{13}+4979q^{12}+3060q^{11}-844q^{10}-4434q^{9}-4462q^{8}-1809q^{7}+1944q^{6}+3828q^{5}+3677q^{4}+1147q^{3}-2236q^{2}-3594q-2831-290q^{-1}+1775q^{-2}+3078q^{-3}+2319q^{-4}+68q^{-5}-1695q^{-6}-2423q^{-7}-1547q^{-8}-320q^{-9}+1362q^{-10}+1925q^{-11}+1247q^{-12}+161q^{-13}-917q^{-14}-1199q^{-15}-1184q^{-16}-195q^{-17}+578q^{-18}+881q^{-19}+759q^{-20}+271q^{-21}-126q^{-22}-684q^{-23}-536q^{-24}-284q^{-25}+48q^{-26}+284q^{-27}+368q^{-28}+360q^{-29}-15q^{-30}-124q^{-31}-236q^{-32}-196q^{-33}-118q^{-34}+28q^{-35}+186q^{-36}+103q^{-37}+89q^{-38}-2q^{-39}-48q^{-40}-94q^{-41}-62q^{-42}+12q^{-43}+10q^{-44}+39q^{-45}+25q^{-46}+17q^{-47}-16q^{-48}-17q^{-49}-q^{-50}-7q^{-51}+3q^{-52}+3q^{-53}+5q^{-54}-q^{-55}-3q^{-56}+q^{-57}$
7	$2q^{131}-4q^{130}-2q^{129}-2q^{128}+15q^{126}+19q^{125}+15q^{124}-13q^{123}-47q^{122}-70q^{121}-68q^{120}-28q^{119}+109q^{118}+246q^{117}+278q^{116}+144q^{115}-160q^{114}-489q^{113}-725q^{112}-649q^{111}-75q^{110}+846q^{109}+1567q^{108}+1653q^{107}+791q^{106}-812q^{105}-2516q^{104}-3461q^{103}-2611q^{102}+49q^{101}+3396q^{100}+5762q^{99}+5505q^{98}+2098q^{97}-3244q^{96}-8154q^{95}-9574q^{94}-5881q^{93}+1670q^{92}+9855q^{91}+13991q^{90}+11087q^{89}+1803q^{88}-9990q^{87}-17999q^{86}-17161q^{85}-6983q^{84}+8363q^{83}+20776q^{82}+22927q^{81}+13091q^{80}-4888q^{79}-21648q^{78}-27727q^{77}-19347q^{76}+351q^{75}+20885q^{74}+30785q^{73}+24610q^{72}+4578q^{71}-18666q^{70}-32143q^{69}-28605q^{68}-8988q^{67}+15921q^{66}+32023q^{65}+30941q^{64}+12461q^{63}-13064q^{62}-31045q^{61}-32019q^{60}-14775q^{59}+10730q^{58}+29642q^{57}+32091q^{56}+16131q^{55}-8934q^{54}-28287q^{53}-31718q^{52}-16723q^{51}+7756q^{50}+27068q^{49}+31078q^{48}+16987q^{47}-6870q^{46}-26075q^{45}-30510q^{44}-17103q^{43}+6142q^{42}+25133q^{41}+29948q^{40}+17342q^{39}-5240q^{38}-24091q^{37}-29454q^{36}-17828q^{35}+4011q^{34}+22811q^{33}+28917q^{32}+18523q^{31}-2320q^{30}-20988q^{29}-28182q^{28}-19533q^{27}+19q^{26}+18678q^{25}+27138q^{24}+20527q^{23}+2746q^{22}-15509q^{21}-25433q^{20}-21539q^{19}-6000q^{18}+11708q^{17}+23066q^{16}+21943q^{15}+9245q^{14}-7100q^{13}-19589q^{12}-21648q^{11}-12361q^{10}+2229q^{9}+15299q^{8}+20080q^{7}+14458q^{6}+2675q^{5}-10035q^{4}-17269q^{3}-15407q^{2}-6824q+4661+13133q^{-1}+14552q^{-2}+9688q^{-3}+496q^{-4}-8266q^{-5}-12190q^{-6}-10696q^{-7}-4404q^{-8}+3309q^{-9}+8477q^{-10}+9890q^{-11}+6668q^{-12}+794q^{-13}-4385q^{-14}-7487q^{-15}-6944q^{-16}-3494q^{-17}+670q^{-18}+4387q^{-19}+5691q^{-20}+4371q^{-21}+1803q^{-22}-1381q^{-23}-3472q^{-24}-3781q^{-25}-2888q^{-26}-721q^{-27}+1299q^{-28}+2355q^{-29}+2579q^{-30}+1629q^{-31}+318q^{-32}-772q^{-33}-1648q^{-34}-1589q^{-35}-970q^{-36}-256q^{-37}+579q^{-38}+907q^{-39}+926q^{-40}+734q^{-41}+138q^{-42}-303q^{-43}-529q^{-44}-605q^{-45}-368q^{-46}-146q^{-47}+94q^{-48}+360q^{-49}+332q^{-50}+236q^{-51}+88q^{-52}-96q^{-53}-140q^{-54}-175q^{-55}-152q^{-56}-30q^{-57}+42q^{-58}+84q^{-59}+90q^{-60}+32q^{-61}+21q^{-62}-12q^{-63}-42q^{-64}-30q^{-65}-18q^{-66}+4q^{-67}+15q^{-68}+4q^{-69}+5q^{-70}+7q^{-71}-3q^{-72}-3q^{-73}-5q^{-74}+q^{-75}+3q^{-76}-q^{-77}$

Computer Talk

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[9, 47]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 8, 17, 7], X[8, 3, 9, 4], X[2, 15, 3, 16], X[14, 9, 15, 10], X[10, 6, 11, 5], X[4, 14, 5, 13], X[11, 1, 12, 18], X[17, 13, 18, 12]]
In[3]:=	GaussCode[Knot[9, 47]]
Out[3]=	GaussCode[1, -4, 3, -7, 6, -1, 2, -3, 5, -6, -8, 9, 7, -5, 4, -2, -9, 8]
In[4]:=	DTCode[Knot[9, 47]]
Out[4]=	DTCode[6, 8, 10, 16, 14, -18, 4, 2, -12]
In[5]:=	br = BR[Knot[9, 47]]
Out[5]=	BR[4, {-1, 2, -1, 2, 3, 2, -1, 2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 47]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 47]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 47]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, {4, 6}, 2}
In[10]:=	alex = Alexander[Knot[9, 47]][t]
Out[10]=	-3 4 6 2 3 -5 + t - -- + - + 6 t - 4 t + t 2 t t
In[11]:=	Conway[Knot[9, 47]][z]
Out[11]=	2 4 6 1 - z + 2 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 47]}
In[13]:=	{KnotDet[Knot[9, 47]], KnotSignature[Knot[9, 47]]}
Out[13]=	{27, 2}
In[14]:=	Jones[Knot[9, 47]][q]
Out[14]=	-2 3 2 3 4 5 -3 - q + - + 5 q - 5 q + 4 q - 4 q + 2 q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 47]}
In[16]:=	A2Invariant[Knot[9, 47]][q]
Out[16]=	-6 -4 -2 2 4 6 8 12 14 16 20 2 - q + q + q + 2 q - q + q - 2 q - q - q + q + q
In[17]:=	HOMFLYPT[Knot[9, 47]][a, z]
Out[17]=	2 2 4 4 6 -6 2 -2 2 3 z 4 z 4 z 4 z z 1 + a - -- + a - 2 z - ---- + ---- - z - -- + ---- + -- 4 4 2 4 2 2 a a a a a a
In[18]:=	Kauffman[Knot[9, 47]][a, z]
Out[18]=	2 2 2 -6 2 -2 3 z 5 z 2 z 2 3 z 9 z 11 z 1 - a - -- - a - --- - --- - --- + 5 z + ---- + ---- + ----- + 4 5 3 a 6 4 2 a a a a a a 3 3 3 4 4 5 5 5 3 z 6 z z 3 4 7 z 16 z z 4 z 4 z ---- + ---- + -- - 2 a z - 9 z - ---- - ----- + -- - ---- - ---- + 5 3 a 4 2 5 3 a a a a a a a 6 6 7 7 5 6 3 z 6 z 2 z 2 z a z + 3 z + ---- + ---- + ---- + ---- 4 2 3 a a a a
In[19]:=	{Vassiliev[2][Knot[9, 47]], Vassiliev[3][Knot[9, 47]]}
Out[19]=	{-1, -2}
In[20]:=	Kh[Knot[9, 47]][q, t]
Out[20]=	3 1 2 1 1 2 q 3 5 4 q + 2 q + ----- + ----- + ---- + --- + --- + 2 q t + 3 q t + 5 3 3 2 2 q t t q t q t q t 5 2 7 2 7 3 9 3 11 4 2 q t + 2 q t + 2 q t + 2 q t + 2 q t
In[21]:=	ColouredJones[Knot[9, 47], 2][q]
Out[21]=	-7 3 -5 9 6 8 17 2 3 4 -3 + q - -- - q + -- - -- - -- + -- - 18 q + 20 q + 2 q - 25 q + 6 4 3 2 q q q q q 5 6 7 8 9 10 11 12 19 q + 8 q - 25 q + 14 q + 10 q - 17 q + 6 q + 6 q - 13 15 6 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

9_46 9_48