# 9 49

Jump to navigationJump to search

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 9 49's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 49 at Knotilus!

### Knot presentations

 Planar diagram presentation X6271 X12,8,13,7 X5,15,6,14 X3,11,4,10 X11,3,12,2 X15,5,16,4 X17,9,18,8 X9,17,10,16 X18,14,1,13 Gauss code 1, 5, -4, 6, -3, -1, 2, 7, -8, 4, -5, -2, 9, 3, -6, 8, -7, -9 Dowker-Thistlethwaite code 6 -10 -14 12 -16 -2 18 -4 -8 Conway Notation [-20:-20:-20]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 11, width is 4,

Braid index is 4

[{2, 7}, {1, 5}, {8, 3}, {7, 9}, {6, 2}, {4, 1}, {5, 8}, {3, 6}, {9, 4}]

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 3 3-genus 2 Bridge index 3 Super bridge index ${\displaystyle \{4,5\}}$ Nakanishi index 2 Maximal Thurston-Bennequin number [3][-12] Hyperbolic Volume 9.42707 A-Polynomial See Data:9 49/A-polynomial

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 2}$ Topological 4 genus ${\displaystyle 2}$ Concordance genus ${\displaystyle 2}$ Rasmussen s-Invariant -4

### Polynomial invariants

 Alexander polynomial ${\displaystyle 3t^{2}-6t+7-6t^{-1}+3t^{-2}}$ Conway polynomial ${\displaystyle 3z^{4}+6z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{5,t+1\}}$ Determinant and Signature { 25, 4 } Jones polynomial ${\displaystyle -2q^{9}+3q^{8}-4q^{7}+5q^{6}-4q^{5}+4q^{4}-2q^{3}+q^{2}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle z^{4}a^{-4}+2z^{4}a^{-6}+2z^{2}a^{-4}+6z^{2}a^{-6}-2z^{2}a^{-8}+4a^{-6}-3a^{-8}}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{7}a^{-7}+z^{7}a^{-9}+3z^{6}a^{-6}+4z^{6}a^{-8}+z^{6}a^{-10}+2z^{5}a^{-5}+z^{5}a^{-7}-z^{5}a^{-9}+z^{4}a^{-4}-8z^{4}a^{-6}-9z^{4}a^{-8}-3z^{3}a^{-5}-3z^{3}a^{-7}+3z^{3}a^{-9}+3z^{3}a^{-11}-2z^{2}a^{-4}+9z^{2}a^{-6}+10z^{2}a^{-8}-z^{2}a^{-10}+2za^{-7}-2za^{-9}-4za^{-11}-4a^{-6}-3a^{-8}}$ The A2 invariant ${\displaystyle q^{-6}-q^{-8}+q^{-10}+q^{-14}+3q^{-16}+q^{-18}+2q^{-20}-q^{-22}-q^{-24}-q^{-26}-2q^{-28}}$ The G2 invariant ${\displaystyle q^{-30}-q^{-32}+2q^{-34}-3q^{-36}+2q^{-38}-q^{-40}-2q^{-42}+8q^{-44}-10q^{-46}+12q^{-48}-7q^{-50}-q^{-52}+10q^{-54}-16q^{-56}+19q^{-58}-11q^{-60}+q^{-62}+10q^{-64}-14q^{-66}+13q^{-68}-2q^{-70}-6q^{-72}+14q^{-74}-12q^{-76}+4q^{-78}+9q^{-80}-15q^{-82}+21q^{-84}-16q^{-86}+9q^{-88}+5q^{-90}-13q^{-92}+22q^{-94}-22q^{-96}+16q^{-98}-4q^{-100}-7q^{-102}+13q^{-104}-16q^{-106}+9q^{-108}-q^{-110}-10q^{-112}+10q^{-114}-11q^{-116}-2q^{-118}+10q^{-120}-20q^{-122}+16q^{-124}-9q^{-126}-4q^{-128}+11q^{-130}-16q^{-132}+15q^{-134}-7q^{-136}+q^{-138}+3q^{-140}-7q^{-142}+7q^{-144}-2q^{-146}+q^{-148}+q^{-150}}$

### "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

### Vassiliev invariants

 V2 and V3: (6, 14)
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 ${\displaystyle 24}$ ${\displaystyle 112}$ ${\displaystyle 288}$ ${\displaystyle 700}$ ${\displaystyle 116}$ ${\displaystyle 2688}$ ${\displaystyle {\frac {14368}{3}}}$ ${\displaystyle {\frac {2560}{3}}}$ ${\displaystyle 688}$ ${\displaystyle 2304}$ ${\displaystyle 6272}$ ${\displaystyle 16800}$ ${\displaystyle 2784}$ ${\displaystyle {\frac {166191}{5}}}$ ${\displaystyle {\frac {1876}{5}}}$ ${\displaystyle {\frac {207244}{15}}}$ ${\displaystyle {\frac {689}{3}}}$ ${\displaystyle {\frac {9391}{5}}}$

V2,1 through V6,9 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 ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$). The squares with yellow highlighting are those on the "critical diagonals", where ${\displaystyle j-2r=s+1}$ or ${\displaystyle j-2r=s-1}$, where ${\displaystyle s=}$4 is the signature of 9 49. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
01234567χ
19       2-2
17      1 1
15     32 -1
13    21  1
11   23   1
9  22    0
7  2     2
512      -1
31       1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=3}$ ${\displaystyle i=5}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$